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Your data matches 14 different statistics following compositions of up to 3 maps.
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Matching statistic: St000319

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2,1] => [[2,2],[1]]
 => [1]
 => 0
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[2,2] => [[3,2],[1]]
 => [1]
 => 0
[3,1] => [[3,3],[2]]
 => [2]
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[2,3] => [[4,2],[1]]
 => [1]
 => 0
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
[3,2] => [[4,3],[2]]
 => [2]
 => 1
[4,1] => [[4,4],[3]]
 => [3]
 => 2
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 0
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 0
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 1
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => 2
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 1
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 2
[2,4] => [[5,2],[1]]
 => [1]
 => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 1
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 2
[3,3] => [[5,3],[2]]
 => [2]
 => 1
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 3
[4,2] => [[5,4],[3]]
 => [3]
 => 2
[5,1] => [[5,5],[4]]
 => [4]
 => 3
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => 0
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => [2,1]
 => 1
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => 0

Description

The spin of an integer partition. The Ferrers shape of an integer partition

$\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of

$\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let

$\lambda = (5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$ The first strip

$(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses

$4$ times, the second strip

$(4,3,3,1) \setminus (2,2)$ crosses

$3$ times, the strip

$(2,2) \setminus (1)$ crosses

$1$ time, and the remaining strip

$(1) \setminus ()$ does not cross. This yields the spin of

$(5,5,4,4,2,1)$ to be

$4+3+1 = 8$ .

Matching statistic: St000320

Mapping

Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[2,1] => [[2,2],[1]]
 => [1]
 => 0
[1,2,1] => [[2,2,1],[1]]
 => [1]
 => 0
[2,1,1] => [[2,2,2],[1,1]]
 => [1,1]
 => 0
[2,2] => [[3,2],[1]]
 => [1]
 => 0
[3,1] => [[3,3],[2]]
 => [2]
 => 1
[1,1,2,1] => [[2,2,1,1],[1]]
 => [1]
 => 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
 => [1,1]
 => 0
[1,2,2] => [[3,2,1],[1]]
 => [1]
 => 0
[1,3,1] => [[3,3,1],[2]]
 => [2]
 => 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
[2,1,2] => [[3,2,2],[1,1]]
 => [1,1]
 => 0
[2,2,1] => [[3,3,2],[2,1]]
 => [2,1]
 => 1
[2,3] => [[4,2],[1]]
 => [1]
 => 0
[3,1,1] => [[3,3,3],[2,2]]
 => [2,2]
 => 1
[3,2] => [[4,3],[2]]
 => [2]
 => 1
[4,1] => [[4,4],[3]]
 => [3]
 => 2
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
 => [1]
 => 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
 => [1,1]
 => 0
[1,1,2,2] => [[3,2,1,1],[1]]
 => [1]
 => 0
[1,1,3,1] => [[3,3,1,1],[2]]
 => [2]
 => 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
 => [1,1,1]
 => 0
[1,2,1,2] => [[3,2,2,1],[1,1]]
 => [1,1]
 => 0
[1,2,2,1] => [[3,3,2,1],[2,1]]
 => [2,1]
 => 1
[1,2,3] => [[4,2,1],[1]]
 => [1]
 => 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
 => [2,2]
 => 1
[1,3,2] => [[4,3,1],[2]]
 => [2]
 => 1
[1,4,1] => [[4,4,1],[3]]
 => [3]
 => 2
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
 => [1,1,1,1]
 => 0
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
 => [1,1,1]
 => 0
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
 => [2,1,1]
 => 1
[2,1,3] => [[4,2,2],[1,1]]
 => [1,1]
 => 0
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
 => [2,2,1]
 => 1
[2,2,2] => [[4,3,2],[2,1]]
 => [2,1]
 => 1
[2,3,1] => [[4,4,2],[3,1]]
 => [3,1]
 => 2
[2,4] => [[5,2],[1]]
 => [1]
 => 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
 => [2,2,2]
 => 1
[3,1,2] => [[4,3,3],[2,2]]
 => [2,2]
 => 1
[3,2,1] => [[4,4,3],[3,2]]
 => [3,2]
 => 2
[3,3] => [[5,3],[2]]
 => [2]
 => 1
[4,1,1] => [[4,4,4],[3,3]]
 => [3,3]
 => 3
[4,2] => [[5,4],[3]]
 => [3]
 => 2
[5,1] => [[5,5],[4]]
 => [4]
 => 3
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
 => [1]
 => 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
 => [1,1]
 => 0
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
 => [1]
 => 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
 => [2]
 => 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
 => [1,1,1]
 => 0
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
 => [1,1]
 => 0
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
 => [2,1]
 => 1
[1,1,2,3] => [[4,2,1,1],[1]]
 => [1]
 => 0

Description

The dinv adjustment of an integer partition. The Ferrers shape of an integer partition

$\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For

$0 \leq j < \lambda_1$ let

$n_j$ be the length of the border strip starting at

$(\lambda_1-j,0)$ . The dinv adjustment is then defined by

$\sum_{j:n_j > 0}(\lambda_1-1-j).$ The following example is taken from Appendix B in [2]: Let

$\lambda=(5,5,4,4,2,1)$ . Removing the border strips successively yields the sequence of partitions

$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$ and we obtain

$(n_0,\ldots,n_4) = (10,7,0,3,1)$ . The dinv adjustment is thus

$4+3+1+0 = 8$ .

Matching statistic: St001727

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
St001727: Permutations ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 50%

Values

[2,1] => [1,1,0,0,1,0]
 => [1,3,2] => [3,1,2] => 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => [1,4,2,3] => 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => [2,4,1,3] => 1 = 0 + 1
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => [3,1,2,4] => 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => [4,1,2,3] => 2 = 1 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => [1,2,5,3,4] => 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => [1,3,5,2,4] => 1 = 0 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => [1,4,2,3,5] => 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => [1,5,2,3,4] => 2 = 1 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => [2,3,5,1,4] => 1 = 0 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => [2,4,1,3,5] => 1 = 0 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => [2,5,1,3,4] => 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => [3,1,2,4,5] => 1 = 0 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => [3,5,1,2,4] => 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => [4,1,2,3,5] => 2 = 1 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => [5,1,2,3,4] => 3 = 2 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,1,6,5] => [1,2,3,6,4,5] => 1 = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,3,1,5,6,4] => [1,2,4,6,3,5] => 1 = 0 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,1,5,4,6] => [1,2,5,3,4,6] => 1 = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [2,3,1,4,6,5] => [1,2,6,3,4,5] => 2 = 1 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,1,4,5,6,3] => [1,3,4,6,2,5] => 1 = 0 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,1,4,5,3,6] => [1,3,5,2,4,6] => 1 = 0 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,6,5] => [1,3,6,2,4,5] => 2 = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,1,4,3,5,6] => [1,4,2,3,5,6] => 1 = 0 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [2,1,3,5,6,4] => [1,4,6,2,3,5] => 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,1,3,5,4,6] => [1,5,2,3,4,6] => 2 = 1 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [2,1,3,4,6,5] => [1,6,2,3,4,5] => 3 = 2 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,3,4,5,6,2] => [2,3,4,6,1,5] => 1 = 0 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,3,4,5,2,6] => [2,3,5,1,4,6] => 1 = 0 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,3,4,2,6,5] => [2,3,6,1,4,5] => 2 = 1 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,3,4,2,5,6] => [2,4,1,3,5,6] => 1 = 0 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,3,2,5,6,4] => [2,4,6,1,3,5] => 2 = 1 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4,6] => [2,5,1,3,4,6] => 2 = 1 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,3,2,4,6,5] => [2,6,1,3,4,5] => 3 = 2 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,3,2,4,5,6] => [3,1,2,4,5,6] => 1 = 0 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,2,4,5,6,3] => [3,4,6,1,2,5] => 2 = 1 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,2,4,5,3,6] => [3,5,1,2,4,6] => 2 = 1 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,2,4,3,6,5] => [3,6,1,2,4,5] => 3 = 2 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,2,4,3,5,6] => [4,1,2,3,5,6] => 2 = 1 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,2,3,5,6,4] => [4,6,1,2,3,5] => 4 = 3 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,2,3,5,4,6] => [5,1,2,3,4,6] => 3 = 2 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,2,3,4,6,5] => [6,1,2,3,4,5] => 4 = 3 + 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,5,1,7,6] => [1,2,3,4,7,5,6] => 1 = 0 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,3,4,1,6,7,5] => [1,2,3,5,7,4,6] => 1 = 0 + 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,4,1,6,5,7] => [1,2,3,6,4,5,7] => 1 = 0 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [2,3,4,1,5,7,6] => [1,2,3,7,4,5,6] => 2 = 1 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,3,1,5,6,7,4] => [1,2,4,5,7,3,6] => 1 = 0 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,3,1,5,6,4,7] => [1,2,4,6,3,5,7] => 1 = 0 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [2,3,1,5,4,7,6] => [1,2,4,7,3,5,6] => 2 = 1 + 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [2,3,1,5,4,6,7] => [1,2,5,3,4,6,7] => 1 = 0 + 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [2,1,3,5,4,7,6] => [1,4,7,2,3,5,6] => ? = 2 + 1
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [2,1,3,5,4,6,7] => [1,5,2,3,4,6,7] => ? = 1 + 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [2,1,3,4,6,7,5] => [1,5,7,2,3,4,6] => ? = 3 + 1
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [2,1,3,4,6,5,7] => [1,6,2,3,4,5,7] => ? = 2 + 1
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [2,1,3,4,5,7,6] => [1,7,2,3,4,5,6] => ? = 3 + 1
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,3,4,5,6,7,2] => [2,3,4,5,7,1,6] => ? = 0 + 1
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,3,4,5,6,2,7] => [2,3,4,6,1,5,7] => ? = 0 + 1
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,3,4,5,2,7,6] => [2,3,4,7,1,5,6] => ? = 1 + 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,3,4,5,2,6,7] => [2,3,5,1,4,6,7] => ? = 0 + 1
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,3,4,2,6,7,5] => [2,3,5,7,1,4,6] => ? = 1 + 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,3,4,2,6,5,7] => [2,3,6,1,4,5,7] => ? = 1 + 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,3,4,2,5,7,6] => [2,3,7,1,4,5,6] => ? = 2 + 1
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,3,4,2,5,6,7] => [2,4,1,3,5,6,7] => ? = 0 + 1
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,3,2,5,6,7,4] => [2,4,5,7,1,3,6] => ? = 1 + 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,3,2,5,6,4,7] => [2,4,6,1,3,5,7] => ? = 1 + 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4,7,6] => [2,4,7,1,3,5,6] => ? = 2 + 1
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,5,4,6,7] => [2,5,1,3,4,6,7] => ? = 1 + 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,3,2,4,6,7,5] => [2,5,7,1,3,4,6] => ? = 3 + 1
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,3,2,4,6,5,7] => [2,6,1,3,4,5,7] => ? = 2 + 1
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,3,2,4,5,7,6] => [2,7,1,3,4,5,6] => ? = 3 + 1
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,3,2,4,5,6,7] => [3,1,2,4,5,6,7] => ? = 0 + 1
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,2,4,5,6,7,3] => [3,4,5,7,1,2,6] => ? = 1 + 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,2,4,5,6,3,7] => [3,4,6,1,2,5,7] => ? = 1 + 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,2,4,5,3,7,6] => [3,4,7,1,2,5,6] => ? = 2 + 1
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,2,4,5,3,6,7] => [3,5,1,2,4,6,7] => ? = 1 + 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,6,7,5] => [3,5,7,1,2,4,6] => ? = 3 + 1
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5,7] => [3,6,1,2,4,5,7] => ? = 2 + 1
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,2,4,3,5,7,6] => [3,7,1,2,4,5,6] => ? = 3 + 1
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,2,4,3,5,6,7] => [4,1,2,3,5,6,7] => ? = 1 + 1
[4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
 => [1,2,3,5,6,7,4] => [4,5,7,1,2,3,6] => ? = 3 + 1
[4,1,2] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
 => [1,2,3,5,6,4,7] => [4,6,1,2,3,5,7] => ? = 3 + 1
[4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
 => [1,2,3,5,4,7,6] => [4,7,1,2,3,5,6] => ? = 4 + 1
[4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => [1,2,3,5,4,6,7] => [5,1,2,3,4,6,7] => ? = 2 + 1
[5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
 => [1,2,3,4,6,7,5] => [5,7,1,2,3,4,6] => ? = 5 + 1
[5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => [1,2,3,4,6,5,7] => [6,1,2,3,4,5,7] => ? = 3 + 1
[6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => [1,2,3,4,5,7,6] => [7,1,2,3,4,5,6] => ? = 4 + 1
[1,1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [2,3,4,5,6,1,8,7] => [1,2,3,4,5,8,6,7] => ? = 0 + 1
[1,1,1,1,2,1,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [2,3,4,5,1,7,8,6] => [1,2,3,4,6,8,5,7] => ? = 0 + 1
[1,1,1,1,2,2] => [1,0,1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [2,3,4,5,1,7,6,8] => [1,2,3,4,7,5,6,8] => ? = 0 + 1
[1,1,1,2,1,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [2,3,4,1,6,7,8,5] => [1,2,3,5,6,8,4,7] => ? = 0 + 1
[1,1,1,2,1,2] => [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [2,3,4,1,6,7,5,8] => [1,2,3,5,7,4,6,8] => ? = 0 + 1
[1,1,1,2,2,1] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [2,3,4,1,6,5,8,7] => [1,2,3,5,8,4,6,7] => ? = 1 + 1
[1,1,1,3,2] => [1,0,1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [2,3,4,1,5,7,6,8] => [1,2,3,7,4,5,6,8] => ? = 1 + 1
[1,1,2,1,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [2,3,1,5,6,7,8,4] => [1,2,4,5,6,8,3,7] => ? = 0 + 1
[1,1,2,1,1,2] => [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [2,3,1,5,6,7,4,8] => [1,2,4,5,7,3,6,8] => ? = 0 + 1
[1,1,2,1,2,1] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,6,4,8,7] => ? => ? = 1 + 1
[1,1,2,2,1,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [2,3,1,5,4,7,8,6] => ? => ? = 1 + 1
[1,1,2,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [2,3,1,5,4,7,6,8] => [1,2,4,7,3,5,6,8] => ? = 1 + 1
[1,1,2,3,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [2,3,1,5,4,6,8,7] => ? => ? = 2 + 1
[1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [2,3,1,4,6,7,5,8] => [1,2,5,7,3,4,6,8] => ? = 1 + 1

Description

The number of invisible inversions of a permutation. A visible inversion of a permutation

$\pi$ is a pair

$i < j$ such that

$\pi(j) \leq \min(i, \pi(i))$ . Thus, an invisible inversion satisfies

$\pi(i) > \pi(j) > i$ .

Matching statistic: St001549

Mapping

Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001549: Permutations ⟶ ℤResult quality: 23% ●values known / values provided: 23%●distinct values known / distinct values provided: 50%

Values

[2,1] => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => [3,1,2] => 0
[1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => [3,1,2,4] => 0
[2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => [4,1,2,3] => 0
[2,2] => [1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => [2,4,1,3] => 0
[3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [3,4,1,2] => 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [3,1,2,4,5] => 0
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [4,1,2,3,5] => 0
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,0,0]
 => [2,4,1,3,5] => 0
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [3,4,1,2,5] => 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [5,1,2,3,4] => 0
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => [2,5,1,3,4] => 0
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [3,5,1,2,4] => 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => [2,3,5,1,4] => 0
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [4,5,1,2,3] => 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => [2,4,5,1,3] => 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [3,4,5,1,2] => 2
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [3,1,2,4,5,6] => 0
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [4,1,2,3,5,6] => 0
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,4,1,3,5,6] => 0
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,1,2,5,6] => 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,0,0,0,0]
 => [5,1,2,3,4,6] => 0
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,5,1,3,4,6] => 0
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4,6] => 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,3,5,1,4,6] => 0
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [4,5,1,2,3,6] => 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,4,5,1,3,6] => 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,1,2,6] => 2
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => [6,1,2,3,4,5] => 0
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,6,1,3,4,5] => 0
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,6,1,2,4,5] => 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,6,1,4,5] => 0
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [4,6,1,2,3,5] => 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,4,6,1,3,5] => 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [3,4,6,1,2,5] => 2
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,6,1,5] => 0
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => [5,6,1,2,3,4] => 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,5,6,1,3,4] => 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [3,5,6,1,2,4] => 2
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,5,6,1,4] => 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,5,6,1,2,3] => 3
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [2,4,5,6,1,3] => 2
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [3,4,5,6,1,2] => 3
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
 => [3,1,2,4,5,6,7] => ? = 0
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
 => [4,1,2,3,5,6,7] => ? = 0
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,4,1,3,5,6,7] => ? = 0
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
 => [3,4,1,2,5,6,7] => ? = 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
 => [5,1,2,3,4,6,7] => ? = 0
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,5,1,3,4,6,7] => ? = 0
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
 => [3,5,1,2,4,6,7] => ? = 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [2,3,5,1,4,6,7] => ? = 0
[1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
 => [4,5,1,2,3,6,7] => ? = 1
[1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,4,5,1,3,6,7] => ? = 1
[1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,1,2,6,7] => ? = 2
[1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
 => [6,1,2,3,4,5,7] => ? = 0
[1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5,7] => ? = 0
[1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
 => [3,6,1,2,4,5,7] => ? = 1
[1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [2,3,6,1,4,5,7] => ? = 0
[1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,1,0,0,0,0,0]
 => [4,6,1,2,3,5,7] => ? = 1
[1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,1,0,0,0,0]
 => [2,4,6,1,3,5,7] => ? = 1
[1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,1,0,0,0,0]
 => [3,4,6,1,2,5,7] => ? = 2
[1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [2,3,4,6,1,5,7] => ? = 0
[1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,0,1,1,0,0,0,0,0]
 => [5,6,1,2,3,4,7] => ? = 1
[1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
 => [2,5,6,1,3,4,7] => ? = 1
[1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,1,1,0,0,0,0]
 => [3,5,6,1,2,4,7] => ? = 2
[1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,6,1,4,7] => ? = 1
[1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
 => [4,5,6,1,2,3,7] => ? = 3
[1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,4,5,6,1,3,7] => ? = 2
[1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
 => [3,4,5,6,1,2,7] => ? = 3
[2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
 => [7,1,2,3,4,5,6] => ? = 0
[2,1,1,1,2] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => [2,7,1,3,4,5,6] => ? = 0
[2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
 => [3,7,1,2,4,5,6] => ? = 1
[2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [2,3,7,1,4,5,6] => ? = 0
[2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,1,0,1,0,0,0,0,0]
 => [4,7,1,2,3,5,6] => ? = 1
[2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,4,7,1,3,5,6] => ? = 1
[2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
 => [3,4,7,1,2,5,6] => ? = 2
[2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [2,3,4,7,1,5,6] => ? = 0
[2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,1,1,0,1,0,0,0,0,0]
 => [5,7,1,2,3,4,6] => ? = 1
[2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
 => [2,5,7,1,3,4,6] => ? = 1
[2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,1,1,0,1,0,0,0,0]
 => [3,5,7,1,2,4,6] => ? = 2
[2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [2,3,5,7,1,4,6] => ? = 1
[2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
 => [4,5,7,1,2,3,6] => ? = 3
[2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,4,5,7,1,3,6] => ? = 2
[2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
 => [3,4,5,7,1,2,6] => ? = 3
[2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [2,3,4,5,7,1,6] => ? = 0
[3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
 => [6,7,1,2,3,4,5] => ? = 1
[3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
 => [2,6,7,1,3,4,5] => ? = 1
[3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,1,0,1,0,1,0,0,0,0]
 => [3,6,7,1,2,4,5] => ? = 2
[3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [2,3,6,7,1,4,5] => ? = 1
[3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
 => [4,6,7,1,2,3,5] => ? = 3
[3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,4,6,7,1,3,5] => ? = 2
[3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
 => [3,4,6,7,1,2,5] => ? = 3
[3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [2,3,4,6,7,1,5] => ? = 1

Description

The number of restricted non-inversions between exceedances. This is for a permutation

$\sigma$ of length

$n$ given by

$\operatorname{nie}(\sigma) = \#\{1 \leq i, j \leq n \mid i < j < \sigma(i) < \sigma(j) \}.$

Matching statistic: St001691

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001691: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 25%

Values

[2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
[2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 0
[3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1
[3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 0
[1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 0
[1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1
[1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 0
[3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
 => ? = 1
[3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2
[3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 1
[4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 3
[4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2
[5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 3
[1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 0
[1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,1,1,3,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => 0
[1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => 1
[1,1,3,2] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[1,1,4,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[1,2,4] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 0
[1,3,1,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
 => ? = 1
[1,3,1,2] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[1,3,3] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[1,4,1,1] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[1,4,2] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[1,5,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[2,1,1,2,1] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[2,1,1,3] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
[2,1,2,1,1] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,5),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[2,1,3,1] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[2,1,4] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 0
[2,2,1,1,1] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,8),(0,9),(1,2),(1,3),(1,7),(1,9),(2,3),(2,6),(2,9),(3,5),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[2,2,3] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[2,3,1,1] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[2,3,2] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[2,4,1] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[2,5] => [1,1,1,1,2,1] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 0
[3,1,1,1,1] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[3,1,1,2] => [1,4,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[3,1,2,1] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[3,1,3] => [1,1,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[3,2,1,1] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[3,2,2] => [1,2,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2

Description

The number of kings in a graph. A vertex of a graph is a king, if all its neighbours have smaller degree. In particular, an isolated vertex is a king.

Matching statistic: St000455
(load all 2 compositions to match this statistic)

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 12% ●values known / values provided: 19%●distinct values known / distinct values provided: 12%

Values

[2,1] => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 0
[1,2,1] => [2,2] => [2,2] => ([(1,3),(2,3)],4)
 => 0
[2,1,1] => [1,3] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 0
[2,2] => [1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[3,1] => [1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1
[1,1,2,1] => [3,2] => [2,3] => ([(2,4),(3,4)],5)
 => 0
[1,2,1,1] => [2,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,2,2] => [2,2,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,3,1] => [2,1,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[2,1,1,1] => [1,4] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 0
[2,1,2] => [1,3,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,2,1] => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[2,3] => [1,2,1,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[3,1,1] => [1,1,3] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[3,2] => [1,1,2,1] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1
[4,1] => [1,1,1,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2
[1,1,1,2,1] => [4,2] => [2,4] => ([(3,5),(4,5)],6)
 => 0
[1,1,2,1,1] => [3,3] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => 0
[1,1,2,2] => [3,2,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,1,3,1] => [3,1,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,2,1,1,1] => [2,4] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => 0
[1,2,1,2] => [2,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,2,2,1] => [2,2,2] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,2,3] => [2,2,1,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,3,1,1] => [2,1,3] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,3,2] => [2,1,2,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[1,4,1] => [2,1,1,2] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[2,1,1,1,1] => [1,5] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 0
[2,1,1,2] => [1,4,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,1,2,1] => [1,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[2,1,3] => [1,3,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,2,1,1] => [1,2,3] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[2,2,2] => [1,2,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[2,3,1] => [1,2,1,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[2,4] => [1,2,1,1,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[3,1,1,1] => [1,1,4] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[3,1,2] => [1,1,3,1] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[3,2,1] => [1,1,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[3,3] => [1,1,2,1,1] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1
[4,1,1] => [1,1,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[4,2] => [1,1,1,2,1] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2
[5,1] => [1,1,1,1,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3
[1,1,1,1,2,1] => [5,2] => [2,5] => ([(4,6),(5,6)],7)
 => 0
[1,1,1,2,1,1] => [4,3] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => 0
[1,1,1,2,2] => [4,2,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,1,1,3,1] => [4,1,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[1,1,2,1,1,1] => [3,4] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => 0
[1,1,2,1,2] => [3,3,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,1,2,2,1] => [3,2,2] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[1,1,2,3] => [3,2,1,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,1,3,1,1] => [3,1,3] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[1,1,3,2] => [3,1,2,1] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[1,1,4,1] => [3,1,1,2] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,2,1,1,1,1] => [2,5] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0
[1,2,1,1,2] => [2,4,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,2,1,2,1] => [2,3,2] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[1,2,1,3] => [2,3,1,1] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,2,2,1,1] => [2,2,3] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[1,2,2,2] => [2,2,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[1,2,3,1] => [2,2,1,2] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,2,4] => [2,2,1,1,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[1,3,1,1,1] => [2,1,4] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[1,3,1,2] => [2,1,3,1] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[1,3,2,1] => [2,1,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,3,3] => [2,1,2,1,1] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[1,4,1,1] => [2,1,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[1,4,2] => [2,1,1,2,1] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[1,5,1] => [2,1,1,1,2] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,1,1,1,1,1] => [1,6] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 0
[2,1,1,1,2] => [1,5,1] => [5,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[2,1,1,2,1] => [1,4,2] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[2,1,1,3] => [1,4,1,1] => [4,1,1,1] => ([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[2,1,2,1,1] => [1,3,3] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[2,1,2,2] => [1,3,2,1] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[2,1,3,1] => [1,3,1,2] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,1,4] => [1,3,1,1,1] => [3,1,1,1,1] => ([(0,3),(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[2,2,1,1,1] => [1,2,4] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[2,2,1,2] => [1,2,3,1] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[2,2,2,1] => [1,2,2,2] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,2,3] => [1,2,2,1,1] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1
[2,3,1,1] => [1,2,1,3] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,3,2] => [1,2,1,2,1] => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2
[2,4,1] => [1,2,1,1,2] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3
[2,5] => [1,2,1,1,1,1] => [2,1,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 0
[3,1,1,1,1] => [1,1,5] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 1

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St001305

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001305: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 25%

Values

[2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
[2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 1
[3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 0
[1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 0
[1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 1
[1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 0
[3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
 => ? = 1
[3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2
[3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 1
[4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 3
[4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2
[5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 3
[1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 0
[1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,1,1,3,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 0
[1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 1
[1,1,3,2] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[1,1,4,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[1,2,4] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 0
[1,3,1,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
 => ? = 1
[1,3,1,2] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[1,3,3] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[1,4,1,1] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[1,4,2] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[1,5,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,1,1,2,1] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[2,1,1,3] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
[2,1,2,1,1] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,5),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[2,1,3,1] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[2,1,4] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 0
[2,2,1,1,1] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,8),(0,9),(1,2),(1,3),(1,7),(1,9),(2,3),(2,6),(2,9),(3,5),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[2,2,3] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[2,3,1,1] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[2,3,2] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[2,4,1] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3

Description

The number of induced cycles on four vertices in a graph.

Matching statistic: St001324

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001324: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 25%

Values

[2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
[2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 1
[3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 0
[1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 0
[1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 1
[1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 0
[3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
 => ? = 1
[3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2
[3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 1
[4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 3
[4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2
[5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 3
[1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 0
[1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,1,1,3,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 0
[1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 1
[1,1,3,2] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[1,1,4,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[1,2,4] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 0
[1,3,1,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
 => ? = 1
[1,3,1,2] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[1,3,3] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[1,4,1,1] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[1,4,2] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[1,5,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,1,1,2,1] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[2,1,1,3] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
[2,1,2,1,1] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,5),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[2,1,3,1] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[2,1,4] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 0
[2,2,1,1,1] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,8),(0,9),(1,2),(1,3),(1,7),(1,9),(2,3),(2,6),(2,9),(3,5),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[2,2,3] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[2,3,1,1] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[2,3,2] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[2,4,1] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3

Description

The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. A graph is chordal if and only if in any linear ordering of its vertices, there are no three vertices

$a < b < c$ such that

$(a,c)$ and

$(b,c)$ are edges and

$(a,b)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

Matching statistic: St001326

Mapping

Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001326: Graphs ⟶ ℤResult quality: 18% ●values known / values provided: 18%●distinct values known / distinct values provided: 25%

Values

[2,1] => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
[2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 0
[3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 1
[3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[1,1,1,2,1] => [2,4] => ([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 0
[1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,1,2,2] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,1,3,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 0
[1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 1
[1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[2,1,1,1,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,1,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[2,1,3] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[2,2,1,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[2,3,1] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[2,4] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 0
[3,1,1,1] => [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
 => ? = 1
[3,1,2] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[3,2,1] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2
[3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 1
[4,1,1] => [3,1,1,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 3
[4,2] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 2
[5,1] => [2,1,1,1,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ?
 => ? = 3
[1,1,1,1,2,1] => [2,5] => ([(4,6),(5,6)],7)
 => ([(0,1)],2)
 => 0
[1,1,1,2,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,1,1,2,2] => [1,2,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,1,1,3,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,2,1,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 0
[1,1,2,1,2] => [1,3,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,1,2,2,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 1
[1,1,2,3] => [1,1,2,3] => ([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ? = 0
[1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ? = 1
[1,1,3,2] => [1,2,1,3] => ([(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[1,1,4,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 2
[1,2,1,1,1,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 0
[1,2,1,1,2] => [1,4,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 1
[1,2,1,3] => [1,1,3,2] => ([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 0
[1,2,2,1,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[1,2,3,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
[1,2,4] => [1,1,1,2,2] => ([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 0
[1,3,1,1,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
 => ? = 1
[1,3,1,2] => [1,3,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[1,3,2,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[1,3,3] => [1,1,2,1,2] => ([(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[1,4,1,1] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[1,4,2] => [1,2,1,1,2] => ([(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[1,5,1] => [2,1,1,1,2] => ([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[2,1,1,1,1,1] => [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 0
[2,1,1,1,2] => [1,5,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 0
[2,1,1,2,1] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 1
[2,1,1,3] => [1,1,4,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
[2,1,2,1,1] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,5),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 1
[2,1,2,2] => [1,2,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,4),(0,5),(0,7),(0,8),(1,4),(1,5),(1,6),(1,9),(2,3),(2,6),(2,7),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,8),(4,9),(5,7),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[2,1,3,1] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[2,1,4] => [1,1,1,3,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 0
[2,2,1,1,1] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,8),(0,9),(1,2),(1,3),(1,7),(1,9),(2,3),(2,6),(2,9),(3,5),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 1
[2,2,1,2] => [1,3,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[2,2,3] => [1,1,2,2,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 1
[2,3,1,1] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
[2,3,2] => [1,2,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
[2,4,1] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3

Description

The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. A graph is an interval graph if and only if in any linear ordering of its vertices, there are no three vertices

$a < b < c$ such that

$(a,c)$ is an edge and

$(a,b)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.

Matching statistic: St001651

Mapping

Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00266: Graphs —connected vertex partitions⟶ Lattices
St001651: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 25%

Values

[2,1] => [1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 0
[1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[2,1,1] => [1,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => 0
[2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 0
[3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
[1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 0
[1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 1
[2,1,1,1] => [1,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 0
[2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1
[2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
 => ? = 0
[3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
[3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
 => ? = 1
[4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 2
[1,1,1,2,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[1,1,2,1,1] => [3,3] => ([(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
[1,1,2,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,22),(1,23),(1,24),(1,25),(1,26),(1,27),(2,11),(2,15),(2,20),(2,21),(2,23),(2,50),(3,10),(3,14),(3,18),(3,19),(3,22),(3,50),(4,13),(4,17),(4,19),(4,21),(4,25),(4,49),(5,12),(5,16),(5,18),(5,20),(5,24),(5,49),(6,14),(6,15),(6,16),(6,17),(6,27),(6,48),(7,10),(7,11),(7,12),(7,13),(7,26),(7,48),(8,9),(8,48),(8,49),(8,50),(9,59),(9,60),(9,61),(10,28),(10,40),(10,41),(10,63),(11,29),(11,42),(11,43),(11,63),(12,30),(12,40),(12,42),(12,64),(13,31),(13,41),(13,43),(13,64),(14,32),(14,44),(14,45),(14,63),(15,33),(15,46),(15,47),(15,63),(16,34),(16,44),(16,46),(16,64),(17,35),(17,45),(17,47),(17,64),(18,36),(18,40),(18,44),(18,62),(19,37),(19,41),(19,45),(19,62),(20,38),(20,42),(20,46),(20,62),(21,39),(21,43),(21,47),(21,62),(22,28),(22,32),(22,36),(22,37),(22,59),(23,29),(23,33),(23,38),(23,39),(23,59),(24,30),(24,34),(24,36),(24,38),(24,60),(25,31),(25,35),(25,37),(25,39),(25,60),(26,28),(26,29),(26,30),(26,31),(26,61),(27,32),(27,33),(27,34),(27,35),(27,61),(28,51),(28,52),(28,66),(29,53),(29,54),(29,66),(30,51),(30,53),(30,67),(31,52),(31,54),(31,67),(32,55),(32,56),(32,66),(33,57),(33,58),(33,66),(34,55),(34,57),(34,67),(35,56),(35,58),(35,67),(36,51),(36,55),(36,65),(37,52),(37,56),(37,65),(38,53),(38,57),(38,65),(39,54),(39,58),(39,65),(40,51),(40,68),(41,52),(41,68),(42,53),(42,68),(43,54),(43,68),(44,55),(44,68),(45,56),(45,68),(46,57),(46,68),(47,58),(47,68),(48,61),(48,63),(48,64),(49,60),(49,62),(49,64),(50,59),(50,62),(50,63),(51,69),(52,69),(53,69),(54,69),(55,69),(56,69),(57,69),(58,69),(59,65),(59,66),(60,65),(60,67),(61,66),(61,67),(62,65),(62,68),(63,66),(63,68),(64,67),(64,68),(65,69),(66,69),(67,69),(68,69)],70)
 => ? = 0
[1,1,3,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
 => ? = 1
[1,2,1,1,1] => [2,4] => ([(3,5),(4,5)],6)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[1,2,1,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(1,9),(1,36),(1,37),(2,12),(2,16),(2,21),(2,22),(2,37),(3,11),(3,15),(3,19),(3,20),(3,37),(4,14),(4,18),(4,20),(4,22),(4,36),(5,13),(5,17),(5,19),(5,21),(5,36),(6,9),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,10),(7,11),(7,12),(7,13),(7,14),(8,35),(8,38),(8,39),(9,35),(9,40),(9,41),(10,31),(10,32),(10,33),(10,34),(10,35),(11,23),(11,24),(11,31),(11,38),(12,25),(12,26),(12,32),(12,38),(13,23),(13,25),(13,33),(13,39),(14,24),(14,26),(14,34),(14,39),(15,27),(15,28),(15,31),(15,40),(16,29),(16,30),(16,32),(16,40),(17,27),(17,29),(17,33),(17,41),(18,28),(18,30),(18,34),(18,41),(19,23),(19,27),(19,48),(20,24),(20,28),(20,48),(21,25),(21,29),(21,48),(22,26),(22,30),(22,48),(23,42),(23,49),(24,43),(24,49),(25,44),(25,49),(26,45),(26,49),(27,42),(27,50),(28,43),(28,50),(29,44),(29,50),(30,45),(30,50),(31,42),(31,43),(31,46),(32,44),(32,45),(32,46),(33,42),(33,44),(33,47),(34,43),(34,45),(34,47),(35,46),(35,47),(36,39),(36,41),(36,48),(37,38),(37,40),(37,48),(38,46),(38,49),(39,47),(39,49),(40,46),(40,50),(41,47),(41,50),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,49),(48,50),(49,51),(50,51)],52)
 => ? = 0
[1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 1
[1,2,3] => [2,2,1,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,16),(1,17),(1,30),(1,31),(1,39),(1,40),(1,41),(1,42),(1,49),(2,14),(2,15),(2,28),(2,29),(2,35),(2,36),(2,37),(2,38),(2,49),(3,19),(3,23),(3,27),(3,34),(3,36),(3,40),(3,74),(3,79),(4,18),(4,22),(4,26),(4,33),(4,35),(4,39),(4,74),(4,78),(5,21),(5,22),(5,24),(5,34),(5,37),(5,41),(5,75),(5,76),(6,20),(6,23),(6,25),(6,33),(6,38),(6,42),(6,75),(6,77),(7,13),(7,20),(7,21),(7,29),(7,31),(7,32),(7,78),(7,79),(8,12),(8,18),(8,19),(8,28),(8,30),(8,32),(8,76),(8,77),(9,14),(9,16),(9,25),(9,26),(9,48),(9,76),(9,79),(10,15),(10,17),(10,24),(10,27),(10,48),(10,77),(10,78),(11,12),(11,13),(11,48),(11,49),(11,74),(11,75),(12,43),(12,84),(12,98),(12,121),(13,43),(13,85),(13,99),(13,122),(14,55),(14,56),(14,86),(14,89),(14,109),(15,54),(15,57),(15,87),(15,88),(15,109),(16,59),(16,60),(16,90),(16,93),(16,109),(17,58),(17,61),(17,91),(17,92),(17,109),(18,62),(18,66),(18,84),(18,94),(18,113),(19,63),(19,67),(19,84),(19,95),(19,114),(20,64),(20,68),(20,85),(20,97),(20,113),(21,65),(21,69),(21,85),(21,96),(21,114),(22,50),(22,52),(22,94),(22,96),(22,110),(23,51),(23,53),(23,95),(23,97),(23,110),(24,54),(24,58),(24,71),(24,96),(24,121),(25,55),(25,59),(25,70),(25,97),(25,121),(26,56),(26,60),(26,70),(26,94),(26,122),(27,57),(27,61),(27,71),(27,95),(27,122),(28,62),(28,63),(28,72),(28,86),(28,87),(28,98),(29,64),(29,65),(29,72),(29,88),(29,89),(29,99),(30,66),(30,67),(30,73),(30,90),(30,91),(30,98),(31,68),(31,69),(31,73),(31,92),(31,93),(31,99),(32,43),(32,72),(32,73),(32,113),(32,114),(33,44),(33,46),(33,70),(33,110),(33,113),(34,45),(34,47),(34,71),(34,110),(34,114),(35,44),(35,50),(35,56),(35,62),(35,88),(35,111),(36,45),(36,51),(36,57),(36,63),(36,89),(36,111),(37,45),(37,50),(37,54),(37,65),(37,86),(37,112),(38,44),(38,51),(38,55),(38,64),(38,87),(38,112),(39,46),(39,52),(39,60),(39,66),(39,92),(39,111),(40,47),(40,53),(40,61),(40,67),(40,93),(40,111),(41,47),(41,52),(41,58),(41,69),(41,90),(41,112),(42,46),(42,53),(42,59),(42,68),(42,91),(42,112),(43,100),(43,126),(44,80),(44,117),(44,123),(45,81),(45,118),(45,123),(46,82),(46,119),(46,123),(47,83),(47,120),(47,123),(48,109),(48,121),(48,122),(49,98),(49,99),(49,109),(49,111),(49,112),(50,101),(50,103),(50,123),(51,102),(51,104),(51,123),(52,105),(52,107),(52,123),(53,106),(53,108),(53,123),(54,81),(54,103),(54,124),(55,80),(55,104),(55,124),(56,80),(56,101),(56,125),(57,81),(57,102),(57,125),(58,83),(58,107),(58,124),(59,82),(59,108),(59,124),(60,82),(60,105),(60,125),(61,83),(61,106),(61,125),(62,101),(62,115),(62,117),(63,102),(63,115),(63,118),(64,104),(64,116),(64,117),(65,103),(65,116),(65,118),(66,105),(66,115),(66,119),(67,106),(67,115),(67,120),(68,108),(68,116),(68,119),(69,107),(69,116),(69,120),(70,80),(70,82),(70,126),(71,81),(71,83),(71,126),(72,100),(72,117),(72,118),(73,100),(73,119),(73,120),(74,84),(74,110),(74,111),(74,122),(75,85),(75,110),(75,112),(75,121),(76,86),(76,90),(76,94),(76,114),(76,121),(77,87),(77,91),(77,95),(77,113),(77,121),(78,88),(78,92),(78,96),(78,113),(78,122),(79,89),(79,93),(79,97),(79,114),(79,122),(80,127),(81,127),(82,127),(83,127),(84,115),(84,126),(85,116),(85,126),(86,101),(86,118),(86,124),(87,102),(87,117),(87,124),(88,103),(88,117),(88,125),(89,104),(89,118),(89,125),(90,105),(90,120),(90,124),(91,106),(91,119),(91,124),(92,107),(92,119),(92,125),(93,108),(93,120),(93,125),(94,101),(94,105),(94,126),(95,102),(95,106),(95,126),(96,103),(96,107),(96,126),(97,104),(97,108),(97,126),(98,100),(98,115),(98,124),(99,100),(99,116),(99,125),(100,127),(101,127),(102,127),(103,127),(104,127),(105,127),(106,127),(107,127),(108,127),(109,124),(109,125),(110,123),(110,126),(111,115),(111,123),(111,125),(112,116),(112,123),(112,124),(113,117),(113,119),(113,126),(114,118),(114,120),(114,126),(115,127),(116,127),(117,127),(118,127),(119,127),(120,127),(121,124),(121,126),(122,125),(122,126),(123,127),(124,127),(125,127),(126,127)],128)
 => ? = 0
[1,3,1,1] => [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 1
[1,3,2] => [2,1,2,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,17),(1,18),(1,19),(1,29),(1,30),(1,31),(1,32),(1,33),(1,34),(2,15),(2,16),(2,22),(2,28),(2,31),(2,55),(2,56),(3,12),(3,14),(3,21),(3,27),(3,30),(3,54),(3,56),(4,11),(4,13),(4,20),(4,26),(4,29),(4,54),(4,55),(5,13),(5,14),(5,25),(5,28),(5,32),(5,57),(5,58),(6,11),(6,15),(6,23),(6,27),(6,33),(6,57),(6,59),(7,12),(7,16),(7,24),(7,26),(7,34),(7,58),(7,59),(8,17),(8,20),(8,24),(8,41),(8,56),(8,57),(9,18),(9,21),(9,23),(9,41),(9,55),(9,58),(10,19),(10,22),(10,25),(10,41),(10,54),(10,59),(11,35),(11,70),(11,74),(11,82),(12,36),(12,71),(12,75),(12,82),(13,37),(13,70),(13,73),(13,83),(14,38),(14,71),(14,73),(14,84),(15,39),(15,72),(15,74),(15,84),(16,40),(16,72),(16,75),(16,83),(17,45),(17,49),(17,63),(17,66),(17,67),(18,46),(18,48),(18,63),(18,65),(18,68),(19,47),(19,50),(19,63),(19,64),(19,69),(20,45),(20,51),(20,70),(20,89),(21,46),(21,52),(21,71),(21,89),(22,47),(22,53),(22,72),(22,89),(23,48),(23,52),(23,74),(23,88),(24,49),(24,51),(24,75),(24,88),(25,50),(25,53),(25,73),(25,88),(26,42),(26,51),(26,82),(26,83),(27,43),(27,52),(27,82),(27,84),(28,44),(28,53),(28,83),(28,84),(29,35),(29,37),(29,42),(29,45),(29,64),(29,65),(30,36),(30,38),(30,43),(30,46),(30,64),(30,66),(31,39),(31,40),(31,44),(31,47),(31,65),(31,66),(32,37),(32,38),(32,44),(32,50),(32,67),(32,68),(33,35),(33,39),(33,43),(33,48),(33,67),(33,69),(34,36),(34,40),(34,42),(34,49),(34,68),(34,69),(35,76),(35,80),(35,85),(36,77),(36,81),(36,85),(37,76),(37,79),(37,86),(38,77),(38,79),(38,87),(39,78),(39,80),(39,87),(40,78),(40,81),(40,86),(41,63),(41,88),(41,89),(42,60),(42,85),(42,86),(43,61),(43,85),(43,87),(44,62),(44,86),(44,87),(45,60),(45,76),(45,90),(46,61),(46,77),(46,90),(47,62),(47,78),(47,90),(48,61),(48,80),(48,91),(49,60),(49,81),(49,91),(50,62),(50,79),(50,91),(51,60),(51,92),(52,61),(52,92),(53,62),(53,92),(54,64),(54,73),(54,82),(54,89),(55,65),(55,74),(55,83),(55,89),(56,66),(56,75),(56,84),(56,89),(57,67),(57,70),(57,84),(57,88),(58,68),(58,71),(58,83),(58,88),(59,69),(59,72),(59,82),(59,88),(60,93),(61,93),(62,93),(63,90),(63,91),(64,79),(64,85),(64,90),(65,80),(65,86),(65,90),(66,81),(66,87),(66,90),(67,76),(67,87),(67,91),(68,77),(68,86),(68,91),(69,78),(69,85),(69,91),(70,76),(70,92),(71,77),(71,92),(72,78),(72,92),(73,79),(73,92),(74,80),(74,92),(75,81),(75,92),(76,93),(77,93),(78,93),(79,93),(80,93),(81,93),(82,85),(82,92),(83,86),(83,92),(84,87),(84,92),(85,93),(86,93),(87,93),(88,91),(88,92),(89,90),(89,92),(90,93),(91,93),(92,93)],94)
 => ? = 1
[1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
 => ? = 2
[2,1,1,1,1] => [1,5] => ([(4,5)],6)
 => ([(0,1)],2)
 => 0
[2,1,1,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,29),(2,13),(2,14),(2,15),(2,29),(3,10),(3,11),(3,12),(3,29),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,19),(7,22),(7,25),(7,28),(8,19),(8,20),(8,23),(8,26),(9,19),(9,21),(9,24),(9,27),(10,20),(10,22),(10,30),(11,21),(11,22),(11,31),(12,20),(12,21),(12,32),(13,23),(13,25),(13,30),(14,24),(14,25),(14,31),(15,23),(15,24),(15,32),(16,26),(16,28),(16,30),(17,27),(17,28),(17,31),(18,26),(18,27),(18,32),(19,33),(19,34),(19,35),(20,33),(20,36),(21,33),(21,37),(22,33),(22,38),(23,34),(23,36),(24,34),(24,37),(25,34),(25,38),(26,35),(26,36),(27,35),(27,37),(28,35),(28,38),(29,30),(29,31),(29,32),(30,36),(30,38),(31,37),(31,38),(32,36),(32,37),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
 => ? = 0
[2,1,2,1] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
 => ? = 1
[2,1,3] => [1,3,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,17),(1,21),(1,25),(1,29),(1,36),(1,37),(1,43),(2,12),(2,16),(2,20),(2,24),(2,28),(2,34),(2,35),(2,43),(3,15),(3,19),(3,23),(3,27),(3,31),(3,35),(3,37),(3,42),(4,14),(4,18),(4,22),(4,26),(4,30),(4,34),(4,36),(4,42),(5,11),(5,12),(5,13),(5,14),(5,15),(5,70),(5,71),(6,20),(6,21),(6,22),(6,23),(6,33),(6,69),(6,71),(7,16),(7,17),(7,18),(7,19),(7,33),(7,68),(7,70),(8,28),(8,29),(8,30),(8,31),(8,32),(8,68),(8,71),(9,24),(9,25),(9,26),(9,27),(9,32),(9,69),(9,70),(10,11),(10,42),(10,43),(10,68),(10,69),(11,80),(11,81),(11,103),(12,38),(12,39),(12,80),(12,82),(12,86),(13,40),(13,41),(13,80),(13,83),(13,87),(14,38),(14,40),(14,81),(14,84),(14,88),(15,39),(15,41),(15,81),(15,85),(15,89),(16,44),(16,45),(16,60),(16,82),(16,99),(17,46),(17,47),(17,61),(17,83),(17,99),(18,44),(18,46),(18,62),(18,84),(18,100),(19,45),(19,47),(19,63),(19,85),(19,100),(20,48),(20,49),(20,60),(20,86),(20,101),(21,50),(21,51),(21,61),(21,87),(21,101),(22,48),(22,50),(22,62),(22,88),(22,102),(23,49),(23,51),(23,63),(23,89),(23,102),(24,52),(24,53),(24,64),(24,82),(24,101),(25,54),(25,55),(25,65),(25,83),(25,101),(26,52),(26,54),(26,66),(26,84),(26,102),(27,53),(27,55),(27,67),(27,85),(27,102),(28,56),(28,57),(28,64),(28,86),(28,99),(29,58),(29,59),(29,65),(29,87),(29,99),(30,56),(30,58),(30,66),(30,88),(30,100),(31,57),(31,59),(31,67),(31,89),(31,100),(32,64),(32,65),(32,66),(32,67),(32,103),(33,60),(33,61),(33,62),(33,63),(33,103),(34,38),(34,44),(34,48),(34,52),(34,56),(34,98),(35,39),(35,45),(35,49),(35,53),(35,57),(35,98),(36,40),(36,46),(36,50),(36,54),(36,58),(36,98),(37,41),(37,47),(37,51),(37,55),(37,59),(37,98),(38,90),(38,94),(38,104),(39,91),(39,95),(39,104),(40,92),(40,96),(40,104),(41,93),(41,97),(41,104),(42,81),(42,98),(42,100),(42,102),(43,80),(43,98),(43,99),(43,101),(44,72),(44,90),(44,105),(45,73),(45,91),(45,105),(46,74),(46,92),(46,105),(47,75),(47,93),(47,105),(48,72),(48,94),(48,106),(49,73),(49,95),(49,106),(50,74),(50,96),(50,106),(51,75),(51,97),(51,106),(52,76),(52,90),(52,106),(53,77),(53,91),(53,106),(54,78),(54,92),(54,106),(55,79),(55,93),(55,106),(56,76),(56,94),(56,105),(57,77),(57,95),(57,105),(58,78),(58,96),(58,105),(59,79),(59,97),(59,105),(60,72),(60,73),(60,107),(61,74),(61,75),(61,107),(62,72),(62,74),(62,108),(63,73),(63,75),(63,108),(64,76),(64,77),(64,107),(65,78),(65,79),(65,107),(66,76),(66,78),(66,108),(67,77),(67,79),(67,108),(68,99),(68,100),(68,103),(69,101),(69,102),(69,103),(70,82),(70,83),(70,84),(70,85),(70,103),(71,86),(71,87),(71,88),(71,89),(71,103),(72,109),(73,109),(74,109),(75,109),(76,109),(77,109),(78,109),(79,109),(80,104),(80,107),(81,104),(81,108),(82,90),(82,91),(82,107),(83,92),(83,93),(83,107),(84,90),(84,92),(84,108),(85,91),(85,93),(85,108),(86,94),(86,95),(86,107),(87,96),(87,97),(87,107),(88,94),(88,96),(88,108),(89,95),(89,97),(89,108),(90,109),(91,109),(92,109),(93,109),(94,109),(95,109),(96,109),(97,109),(98,104),(98,105),(98,106),(99,105),(99,107),(100,105),(100,108),(101,106),(101,107),(102,106),(102,108),(103,107),(103,108),(104,109),(105,109),(106,109),(107,109),(108,109)],110)
 => ? = 0
[2,2,1,1] => [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
 => ? = 1
[2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,10),(1,22),(1,23),(1,24),(1,25),(1,26),(1,29),(1,30),(2,13),(2,18),(2,19),(2,20),(2,21),(2,30),(2,36),(3,12),(3,14),(3,15),(3,16),(3,17),(3,29),(3,36),(4,11),(4,12),(4,13),(4,26),(4,51),(4,52),(5,15),(5,19),(5,23),(5,28),(5,50),(5,52),(6,14),(6,18),(6,22),(6,28),(6,49),(6,51),(7,17),(7,20),(7,24),(7,27),(7,49),(7,52),(8,16),(8,21),(8,25),(8,27),(8,50),(8,51),(9,10),(9,11),(9,36),(9,49),(9,50),(10,31),(10,58),(10,59),(10,60),(11,31),(11,57),(11,74),(12,32),(12,57),(12,63),(12,64),(13,33),(13,57),(13,65),(13,66),(14,37),(14,45),(14,63),(14,72),(15,38),(15,45),(15,64),(15,73),(16,39),(16,46),(16,63),(16,73),(17,40),(17,46),(17,64),(17,72),(18,41),(18,47),(18,65),(18,72),(19,42),(19,47),(19,66),(19,73),(20,43),(20,48),(20,66),(20,72),(21,44),(21,48),(21,65),(21,73),(22,34),(22,37),(22,41),(22,58),(22,61),(23,34),(23,38),(23,42),(23,59),(23,62),(24,35),(24,40),(24,43),(24,58),(24,62),(25,35),(25,39),(25,44),(25,59),(25,61),(26,31),(26,32),(26,33),(26,61),(26,62),(27,35),(27,46),(27,48),(27,74),(28,34),(28,45),(28,47),(28,74),(29,32),(29,37),(29,38),(29,39),(29,40),(29,60),(30,33),(30,41),(30,42),(30,43),(30,44),(30,60),(31,71),(31,77),(32,67),(32,68),(32,71),(33,69),(33,70),(33,71),(34,53),(34,55),(34,77),(35,54),(35,56),(35,77),(36,57),(36,60),(36,72),(36,73),(37,53),(37,67),(37,75),(38,53),(38,68),(38,76),(39,54),(39,67),(39,76),(40,54),(40,68),(40,75),(41,55),(41,69),(41,75),(42,55),(42,70),(42,76),(43,56),(43,70),(43,75),(44,56),(44,69),(44,76),(45,53),(45,78),(46,54),(46,78),(47,55),(47,78),(48,56),(48,78),(49,58),(49,72),(49,74),(50,59),(50,73),(50,74),(51,61),(51,63),(51,65),(51,74),(52,62),(52,64),(52,66),(52,74),(53,79),(54,79),(55,79),(56,79),(57,71),(57,78),(58,75),(58,77),(59,76),(59,77),(60,71),(60,75),(60,76),(61,67),(61,69),(61,77),(62,68),(62,70),(62,77),(63,67),(63,78),(64,68),(64,78),(65,69),(65,78),(66,70),(66,78),(67,79),(68,79),(69,79),(70,79),(71,79),(72,75),(72,78),(73,76),(73,78),(74,77),(74,78),(75,79),(76,79),(77,79),(78,79)],80)
 => ? = 1
[2,3,1] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,26),(1,27),(1,28),(2,9),(2,10),(2,11),(2,29),(2,30),(3,13),(3,17),(3,21),(3,28),(3,30),(4,12),(4,16),(4,21),(4,27),(4,29),(5,15),(5,18),(5,20),(5,27),(5,30),(6,14),(6,19),(6,20),(6,28),(6,29),(7,11),(7,16),(7,17),(7,18),(7,19),(7,26),(8,10),(8,12),(8,13),(8,14),(8,15),(8,26),(9,35),(9,38),(10,31),(10,32),(10,35),(11,33),(11,34),(11,35),(12,22),(12,31),(12,36),(13,22),(13,32),(13,37),(14,23),(14,31),(14,37),(15,23),(15,32),(15,36),(16,24),(16,33),(16,36),(17,24),(17,34),(17,37),(18,25),(18,34),(18,36),(19,25),(19,33),(19,37),(20,23),(20,25),(20,38),(21,22),(21,24),(21,38),(22,39),(23,39),(24,39),(25,39),(26,35),(26,36),(26,37),(27,36),(27,38),(28,37),(28,38),(29,31),(29,33),(29,38),(30,32),(30,34),(30,38),(31,39),(32,39),(33,39),(34,39),(35,39),(36,39),(37,39),(38,39)],40)
 => ? = 2
[2,4] => [1,2,1,1,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(1,25),(1,30),(1,31),(1,36),(1,37),(1,40),(1,43),(1,46),(1,64),(1,65),(2,24),(2,27),(2,29),(2,33),(2,35),(2,39),(2,42),(2,45),(2,63),(2,65),(3,23),(3,26),(3,28),(3,32),(3,34),(3,38),(3,41),(3,44),(3,63),(3,64),(4,20),(4,21),(4,22),(4,23),(4,24),(4,25),(4,90),(4,91),(4,92),(5,16),(5,28),(5,29),(5,43),(5,47),(5,48),(5,84),(5,85),(5,90),(6,14),(6,26),(6,30),(6,42),(6,49),(6,51),(6,84),(6,86),(6,91),(7,15),(7,27),(7,31),(7,41),(7,50),(7,52),(7,85),(7,86),(7,92),(8,19),(8,34),(8,35),(8,46),(8,49),(8,50),(8,87),(8,88),(8,90),(9,17),(9,32),(9,36),(9,45),(9,47),(9,52),(9,87),(9,89),(9,91),(10,18),(10,33),(10,37),(10,44),(10,48),(10,51),(10,88),(10,89),(10,92),(11,15),(11,18),(11,20),(11,38),(11,62),(11,65),(11,84),(11,87),(12,14),(12,17),(12,21),(12,39),(12,62),(12,64),(12,85),(12,88),(13,16),(13,19),(13,22),(13,40),(13,62),(13,63),(13,86),(13,89),(14,67),(14,94),(14,97),(14,112),(14,146),(15,66),(15,93),(15,98),(15,113),(15,146),(16,68),(16,95),(16,96),(16,114),(16,146),(17,70),(17,94),(17,100),(17,116),(17,145),(18,69),(18,93),(18,101),(18,115),(18,145),(19,71),(19,95),(19,99),(19,117),(19,145),(20,59),(20,93),(20,102),(20,111),(20,148),(21,60),(21,94),(21,102),(21,110),(21,149),(22,61),(22,95),(22,102),(22,109),(22,150),(23,59),(23,103),(23,105),(23,109),(23,110),(23,124),(24,60),(24,104),(24,106),(24,109),(24,111),(24,125),(25,61),(25,107),(25,108),(25,110),(25,111),(25,126),(26,53),(26,80),(26,97),(26,103),(26,118),(26,139),(27,54),(27,81),(27,98),(27,104),(27,119),(27,139),(28,55),(28,78),(28,96),(28,105),(28,118),(28,140),(29,56),(29,79),(29,96),(29,106),(29,119),(29,141),(30,57),(30,83),(30,97),(30,107),(30,120),(30,141),(31,58),(31,82),(31,98),(31,108),(31,120),(31,140),(32,55),(32,74),(32,100),(32,103),(32,121),(32,142),(33,56),(33,75),(33,101),(33,104),(33,122),(33,142),(34,53),(34,72),(34,99),(34,105),(34,121),(34,143),(35,54),(35,73),(35,99),(35,106),(35,122),(35,144),(36,58),(36,77),(36,100),(36,107),(36,123),(36,144),(37,57),(37,76),(37,101),(37,108),(37,123),(37,143),(38,59),(38,66),(38,69),(38,118),(38,121),(38,147),(39,60),(39,67),(39,70),(39,119),(39,122),(39,147),(40,61),(40,68),(40,71),(40,120),(40,123),(40,147),(41,66),(41,72),(41,74),(41,124),(41,139),(41,140),(42,67),(42,73),(42,75),(42,125),(42,139),(42,141),(43,68),(43,76),(43,77),(43,126),(43,140),(43,141),(44,69),(44,78),(44,80),(44,124),(44,142),(44,143),(45,70),(45,79),(45,81),(45,125),(45,142),(45,144),(46,71),(46,82),(46,83),(46,126),(46,143),(46,144),(47,55),(47,77),(47,79),(47,114),(47,116),(47,148),(48,56),(48,76),(48,78),(48,114),(48,115),(48,149),(49,53),(49,73),(49,83),(49,112),(49,117),(49,148),(50,54),(50,72),(50,82),(50,113),(50,117),(50,149),(51,57),(51,75),(51,80),(51,112),(51,115),(51,150),(52,58),(52,74),(52,81),(52,113),(52,116),(52,150),(53,152),(53,154),(53,158),(54,153),(54,154),(54,159),(55,151),(55,155),(55,158),(56,151),(56,156),(56,159),(57,152),(57,156),(57,160),(58,153),(58,155),(58,160),(59,127),(59,157),(59,158),(60,128),(60,157),(60,159),(61,129),(61,157),(61,160),(62,102),(62,145),(62,146),(62,147),(63,96),(63,99),(63,109),(63,139),(63,142),(63,147),(64,97),(64,100),(64,110),(64,140),(64,143),(64,147),(65,98),(65,101),(65,111),(65,141),(65,144),(65,147),(66,127),(66,130),(66,164),(67,128),(67,131),(67,164),(68,129),(68,132),(68,164),(69,127),(69,133),(69,165),(70,128),(70,134),(70,165),(71,129),(71,135),(71,165),(72,130),(72,154),(72,162),(73,131),(73,154),(73,163),(74,130),(74,155),(74,161),(75,131),(75,156),(75,161),(76,132),(76,156),(76,162),(77,132),(77,155),(77,163),(78,133),(78,151),(78,162),(79,134),(79,151),(79,163),(80,133),(80,152),(80,161),(81,134),(81,153),(81,161),(82,135),(82,153),(82,162),(83,135),(83,152),(83,163),(84,115),(84,118),(84,141),(84,146),(84,148),(85,116),(85,119),(85,140),(85,146),(85,149),(86,117),(86,120),(86,139),(86,146),(86,150),(87,113),(87,121),(87,144),(87,145),(87,148),(88,112),(88,122),(88,143),(88,145),(88,149),(89,114),(89,123),(89,142),(89,145),(89,150),(90,95),(90,105),(90,106),(90,126),(90,148),(90,149),(91,94),(91,103),(91,107),(91,125),(91,148),(91,150),(92,93),(92,104),(92,108),(92,124),(92,149),(92,150),(93,127),(93,138),(93,166),(94,128),(94,137),(94,166),(95,129),(95,136),(95,166),(96,136),(96,151),(96,164),(97,137),(97,152),(97,164),(98,138),(98,153),(98,164),(99,136),(99,154),(99,165),(100,137),(100,155),(100,165),(101,138),(101,156),(101,165),(102,157),(102,166),(103,137),(103,158),(103,161),(104,138),(104,159),(104,161),(105,136),(105,158),(105,162),(106,136),(106,159),(106,163),(107,137),(107,160),(107,163),(108,138),(108,160),(108,162),(109,136),(109,157),(109,161),(110,137),(110,157),(110,162),(111,138),(111,157),(111,163),(112,131),(112,152),(112,166),(113,130),(113,153),(113,166),(114,132),(114,151),(114,166),(115,133),(115,156),(115,166),(116,134),(116,155),(116,166),(117,135),(117,154),(117,166),(118,133),(118,158),(118,164),(119,134),(119,159),(119,164),(120,135),(120,160),(120,164),(121,130),(121,158),(121,165),(122,131),(122,159),(122,165),(123,132),(123,160),(123,165),(124,127),(124,161),(124,162),(125,128),(125,161),(125,163),(126,129),(126,162),(126,163),(127,167),(128,167),(129,167),(130,167),(131,167),(132,167),(133,167),(134,167),(135,167),(136,167),(137,167),(138,167),(139,154),(139,161),(139,164),(140,155),(140,162),(140,164),(141,156),(141,163),(141,164),(142,151),(142,161),(142,165),(143,152),(143,162),(143,165),(144,153),(144,163),(144,165),(145,165),(145,166),(146,164),(146,166),(147,157),(147,164),(147,165),(148,158),(148,163),(148,166),(149,159),(149,162),(149,166),(150,160),(150,161),(150,166),(151,167),(152,167),(153,167),(154,167),(155,167),(156,167),(157,167),(158,167),(159,167),(160,167),(161,167),(162,167),(163,167),(164,167),(165,167),(166,167)],168)
 => ? = 0
[3,1,1,1] => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1
[3,1,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(1,17),(1,24),(1,38),(1,40),(2,10),(2,16),(2,24),(2,37),(2,39),(3,12),(3,18),(3,23),(3,37),(3,40),(4,13),(4,19),(4,23),(4,38),(4,39),(5,15),(5,21),(5,22),(5,39),(5,40),(6,14),(6,20),(6,22),(6,37),(6,38),(7,9),(7,16),(7,17),(7,18),(7,19),(7,20),(7,21),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(9,31),(9,32),(9,33),(9,34),(9,35),(9,36),(10,25),(10,31),(10,41),(10,43),(11,25),(11,32),(11,42),(11,44),(12,26),(12,33),(12,41),(12,44),(13,26),(13,34),(13,42),(13,43),(14,27),(14,35),(14,41),(14,42),(15,27),(15,36),(15,43),(15,44),(16,28),(16,31),(16,45),(16,47),(17,28),(17,32),(17,46),(17,48),(18,29),(18,33),(18,45),(18,48),(19,29),(19,34),(19,46),(19,47),(20,30),(20,35),(20,45),(20,46),(21,30),(21,36),(21,47),(21,48),(22,27),(22,30),(22,56),(23,26),(23,29),(23,56),(24,25),(24,28),(24,56),(25,49),(25,57),(26,50),(26,57),(27,51),(27,57),(28,49),(28,58),(29,50),(29,58),(30,51),(30,58),(31,49),(31,52),(31,54),(32,49),(32,53),(32,55),(33,50),(33,52),(33,55),(34,50),(34,53),(34,54),(35,51),(35,52),(35,53),(36,51),(36,54),(36,55),(37,41),(37,45),(37,56),(38,42),(38,46),(38,56),(39,43),(39,47),(39,56),(40,44),(40,48),(40,56),(41,52),(41,57),(42,53),(42,57),(43,54),(43,57),(44,55),(44,57),(45,52),(45,58),(46,53),(46,58),(47,54),(47,58),(48,55),(48,58),(49,59),(50,59),(51,59),(52,59),(53,59),(54,59),(55,59),(56,57),(56,58),(57,59),(58,59)],60)
 => ? = 1
[3,2,1] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
 => ? = 2
[3,3] => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,19),(1,20),(1,21),(1,40),(1,41),(1,42),(1,43),(1,44),(1,45),(1,46),(2,16),(2,17),(2,18),(2,34),(2,35),(2,36),(2,37),(2,38),(2,39),(2,46),(3,15),(3,18),(3,21),(3,30),(3,33),(3,76),(3,79),(3,80),(4,14),(4,17),(4,20),(4,29),(4,32),(4,75),(4,78),(4,80),(5,13),(5,16),(5,19),(5,28),(5,31),(5,74),(5,77),(5,80),(6,23),(6,26),(6,28),(6,34),(6,40),(6,71),(6,75),(6,76),(7,22),(7,27),(7,29),(7,35),(7,41),(7,72),(7,74),(7,76),(8,24),(8,25),(8,30),(8,36),(8,42),(8,73),(8,74),(8,75),(9,25),(9,27),(9,31),(9,37),(9,43),(9,71),(9,78),(9,79),(10,24),(10,26),(10,32),(10,38),(10,44),(10,72),(10,77),(10,79),(11,22),(11,23),(11,33),(11,39),(11,45),(11,73),(11,77),(11,78),(12,13),(12,14),(12,15),(12,46),(12,71),(12,72),(12,73),(13,81),(13,84),(13,99),(13,127),(14,81),(14,85),(14,100),(14,128),(15,81),(15,86),(15,101),(15,129),(16,47),(16,50),(16,82),(16,84),(16,87),(16,90),(17,48),(17,51),(17,82),(17,85),(17,88),(17,91),(18,49),(18,52),(18,82),(18,86),(18,89),(18,92),(19,53),(19,56),(19,83),(19,84),(19,93),(19,96),(20,54),(20,57),(20,83),(20,85),(20,94),(20,97),(21,55),(21,58),(21,83),(21,86),(21,95),(21,98),(22,59),(22,65),(22,103),(22,107),(22,127),(23,60),(23,66),(23,102),(23,107),(23,128),(24,61),(24,67),(24,104),(24,106),(24,127),(25,62),(25,68),(25,104),(25,105),(25,128),(26,63),(26,69),(26,102),(26,106),(26,129),(27,64),(27,70),(27,103),(27,105),(27,129),(28,47),(28,53),(28,99),(28,102),(28,131),(29,48),(29,54),(29,100),(29,103),(29,131),(30,49),(30,55),(30,101),(30,104),(30,131),(31,50),(31,56),(31,99),(31,105),(31,130),(32,51),(32,57),(32,100),(32,106),(32,130),(33,52),(33,58),(33,101),(33,107),(33,130),(34,47),(34,60),(34,63),(34,88),(34,89),(34,120),(35,48),(35,59),(35,64),(35,87),(35,89),(35,121),(36,49),(36,61),(36,62),(36,87),(36,88),(36,122),(37,50),(37,62),(37,64),(37,91),(37,92),(37,120),(38,51),(38,61),(38,63),(38,90),(38,92),(38,121),(39,52),(39,59),(39,60),(39,90),(39,91),(39,122),(40,53),(40,66),(40,69),(40,94),(40,95),(40,120),(41,54),(41,65),(41,70),(41,93),(41,95),(41,121),(42,55),(42,67),(42,68),(42,93),(42,94),(42,122),(43,56),(43,68),(43,70),(43,97),(43,98),(43,120),(44,57),(44,67),(44,69),(44,96),(44,98),(44,121),(45,58),(45,65),(45,66),(45,96),(45,97),(45,122),(46,84),(46,85),(46,86),(46,120),(46,121),(46,122),(47,108),(47,124),(47,132),(48,109),(48,125),(48,132),(49,110),(49,126),(49,132),(50,111),(50,124),(50,133),(51,112),(51,125),(51,133),(52,113),(52,126),(52,133),(53,114),(53,124),(53,134),(54,115),(54,125),(54,134),(55,116),(55,126),(55,134),(56,117),(56,124),(56,135),(57,118),(57,125),(57,135),(58,119),(58,126),(58,135),(59,109),(59,113),(59,136),(60,108),(60,113),(60,137),(61,110),(61,112),(61,136),(62,110),(62,111),(62,137),(63,108),(63,112),(63,138),(64,109),(64,111),(64,138),(65,115),(65,119),(65,136),(66,114),(66,119),(66,137),(67,116),(67,118),(67,136),(68,116),(68,117),(68,137),(69,114),(69,118),(69,138),(70,115),(70,117),(70,138),(71,99),(71,120),(71,128),(71,129),(72,100),(72,121),(72,127),(72,129),(73,101),(73,122),(73,127),(73,128),(74,87),(74,93),(74,105),(74,127),(74,131),(75,88),(75,94),(75,106),(75,128),(75,131),(76,89),(76,95),(76,107),(76,129),(76,131),(77,90),(77,96),(77,102),(77,127),(77,130),(78,91),(78,97),(78,103),(78,128),(78,130),(79,92),(79,98),(79,104),(79,129),(79,130),(80,81),(80,82),(80,83),(80,130),(80,131),(81,123),(81,139),(82,123),(82,132),(82,133),(83,123),(83,134),(83,135),(84,123),(84,124),(84,136),(85,123),(85,125),(85,137),(86,123),(86,126),(86,138),(87,111),(87,132),(87,136),(88,112),(88,132),(88,137),(89,113),(89,132),(89,138),(90,108),(90,133),(90,136),(91,109),(91,133),(91,137),(92,110),(92,133),(92,138),(93,117),(93,134),(93,136),(94,118),(94,134),(94,137),(95,119),(95,134),(95,138),(96,114),(96,135),(96,136),(97,115),(97,135),(97,137),(98,116),(98,135),(98,138),(99,124),(99,139),(100,125),(100,139),(101,126),(101,139),(102,108),(102,114),(102,139),(103,109),(103,115),(103,139),(104,110),(104,116),(104,139),(105,111),(105,117),(105,139),(106,112),(106,118),(106,139),(107,113),(107,119),(107,139),(108,140),(109,140),(110,140),(111,140),(112,140),(113,140),(114,140),(115,140),(116,140),(117,140),(118,140),(119,140),(120,124),(120,137),(120,138),(121,125),(121,136),(121,138),(122,126),(122,136),(122,137),(123,140),(124,140),(125,140),(126,140),(127,136),(127,139),(128,137),(128,139),(129,138),(129,139),(130,133),(130,135),(130,139),(131,132),(131,134),(131,139),(132,140),(133,140),(134,140),(135,140),(136,140),(137,140),(138,140),(139,140)],141)
 => ? = 1
[4,1,1] => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,9),(1,11),(1,13),(2,9),(2,10),(2,12),(3,8),(3,10),(3,13),(4,8),(4,11),(4,12),(5,7),(5,12),(5,13),(6,7),(6,10),(6,11),(7,14),(8,14),(9,14),(10,14),(11,14),(12,14),(13,14)],15)
 => ? = 3
[4,2] => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,13),(1,24),(1,25),(1,31),(1,54),(1,55),(1,57),(2,12),(2,22),(2,23),(2,30),(2,52),(2,53),(2,57),(3,15),(3,27),(3,29),(3,33),(3,53),(3,55),(3,56),(4,14),(4,26),(4,28),(4,32),(4,52),(4,54),(4,56),(5,16),(5,22),(5,26),(5,35),(5,55),(5,58),(5,60),(6,17),(6,23),(6,27),(6,36),(6,54),(6,58),(6,61),(7,18),(7,24),(7,28),(7,36),(7,53),(7,59),(7,60),(8,19),(8,25),(8,29),(8,35),(8,52),(8,59),(8,61),(9,21),(9,32),(9,33),(9,34),(9,57),(9,60),(9,61),(10,20),(10,30),(10,31),(10,34),(10,56),(10,58),(10,59),(11,12),(11,13),(11,14),(11,15),(11,16),(11,17),(11,18),(11,19),(11,20),(11,21),(12,37),(12,38),(12,45),(12,72),(12,73),(12,77),(13,39),(13,40),(13,46),(13,74),(13,75),(13,77),(14,41),(14,43),(14,47),(14,72),(14,74),(14,76),(15,42),(15,44),(15,48),(15,73),(15,75),(15,76),(16,37),(16,41),(16,49),(16,75),(16,78),(16,80),(17,38),(17,42),(17,50),(17,74),(17,78),(17,81),(18,39),(18,43),(18,50),(18,73),(18,79),(18,80),(19,40),(19,44),(19,49),(19,72),(19,79),(19,81),(20,45),(20,46),(20,51),(20,76),(20,78),(20,79),(21,47),(21,48),(21,51),(21,77),(21,80),(21,81),(22,37),(22,62),(22,66),(22,96),(23,38),(23,62),(23,67),(23,95),(24,39),(24,63),(24,68),(24,96),(25,40),(25,63),(25,69),(25,95),(26,41),(26,64),(26,66),(26,94),(27,42),(27,65),(27,67),(27,94),(28,43),(28,64),(28,68),(28,93),(29,44),(29,65),(29,69),(29,93),(30,45),(30,62),(30,70),(30,93),(31,46),(31,63),(31,70),(31,94),(32,47),(32,64),(32,71),(32,95),(33,48),(33,65),(33,71),(33,96),(34,51),(34,70),(34,71),(34,92),(35,49),(35,66),(35,69),(35,92),(36,50),(36,67),(36,68),(36,92),(37,82),(37,86),(37,100),(38,82),(38,87),(38,99),(39,83),(39,88),(39,100),(40,83),(40,89),(40,99),(41,84),(41,86),(41,98),(42,85),(42,87),(42,98),(43,84),(43,88),(43,97),(44,85),(44,89),(44,97),(45,82),(45,90),(45,97),(46,83),(46,90),(46,98),(47,84),(47,91),(47,99),(48,85),(48,91),(48,100),(49,86),(49,89),(49,101),(50,87),(50,88),(50,101),(51,90),(51,91),(51,101),(52,66),(52,72),(52,93),(52,95),(53,67),(53,73),(53,93),(53,96),(54,68),(54,74),(54,94),(54,95),(55,69),(55,75),(55,94),(55,96),(56,71),(56,76),(56,93),(56,94),(57,70),(57,77),(57,95),(57,96),(58,62),(58,78),(58,92),(58,94),(59,63),(59,79),(59,92),(59,93),(60,64),(60,80),(60,92),(60,96),(61,65),(61,81),(61,92),(61,95),(62,82),(62,102),(63,83),(63,102),(64,84),(64,102),(65,85),(65,102),(66,86),(66,102),(67,87),(67,102),(68,88),(68,102),(69,89),(69,102),(70,90),(70,102),(71,91),(71,102),(72,86),(72,97),(72,99),(73,87),(73,97),(73,100),(74,88),(74,98),(74,99),(75,89),(75,98),(75,100),(76,91),(76,97),(76,98),(77,90),(77,99),(77,100),(78,82),(78,98),(78,101),(79,83),(79,97),(79,101),(80,84),(80,100),(80,101),(81,85),(81,99),(81,101),(82,103),(83,103),(84,103),(85,103),(86,103),(87,103),(88,103),(89,103),(90,103),(91,103),(92,101),(92,102),(93,97),(93,102),(94,98),(94,102),(95,99),(95,102),(96,100),(96,102),(97,103),(98,103),(99,103),(100,103),(101,103),(102,103)],104)
 => ? = 2
[5,1] => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,13),(1,14),(1,20),(1,28),(1,29),(1,31),(2,11),(2,12),(2,19),(2,26),(2,27),(2,31),(3,16),(3,18),(3,22),(3,27),(3,29),(3,30),(4,15),(4,17),(4,21),(4,26),(4,28),(4,30),(5,11),(5,15),(5,24),(5,29),(5,32),(5,34),(6,12),(6,16),(6,25),(6,28),(6,32),(6,35),(7,13),(7,17),(7,25),(7,27),(7,33),(7,34),(8,14),(8,18),(8,24),(8,26),(8,33),(8,35),(9,21),(9,22),(9,23),(9,31),(9,34),(9,35),(10,19),(10,20),(10,23),(10,30),(10,32),(10,33),(11,36),(11,40),(11,50),(12,36),(12,41),(12,49),(13,37),(13,42),(13,50),(14,37),(14,43),(14,49),(15,38),(15,40),(15,48),(16,39),(16,41),(16,48),(17,38),(17,42),(17,47),(18,39),(18,43),(18,47),(19,36),(19,44),(19,47),(20,37),(20,44),(20,48),(21,38),(21,45),(21,49),(22,39),(22,45),(22,50),(23,44),(23,45),(23,46),(24,40),(24,43),(24,46),(25,41),(25,42),(25,46),(26,40),(26,47),(26,49),(27,41),(27,47),(27,50),(28,42),(28,48),(28,49),(29,43),(29,48),(29,50),(30,45),(30,47),(30,48),(31,44),(31,49),(31,50),(32,36),(32,46),(32,48),(33,37),(33,46),(33,47),(34,38),(34,46),(34,50),(35,39),(35,46),(35,49),(36,51),(37,51),(38,51),(39,51),(40,51),(41,51),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,51),(49,51),(50,51)],52)
 => ? = 3
[1,1,1,1,2,1] => [5,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(1,13),(1,14),(1,15),(2,9),(2,10),(2,11),(2,15),(3,7),(3,8),(3,11),(3,14),(4,6),(4,8),(4,10),(4,13),(5,6),(5,7),(5,9),(5,12),(6,16),(6,19),(6,22),(7,16),(7,17),(7,20),(8,16),(8,18),(8,21),(9,17),(9,19),(9,23),(10,18),(10,19),(10,24),(11,17),(11,18),(11,25),(12,20),(12,22),(12,23),(13,21),(13,22),(13,24),(14,20),(14,21),(14,25),(15,23),(15,24),(15,25),(16,29),(16,30),(17,26),(17,30),(18,27),(18,30),(19,28),(19,30),(20,26),(20,29),(21,27),(21,29),(22,28),(22,29),(23,26),(23,28),(24,27),(24,28),(25,26),(25,27),(26,31),(27,31),(28,31),(29,31),(30,31)],32)
 => 0
[1,1,1,2,1,1] => [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
 => 0
[1,1,1,2,2] => [4,2,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,11),(1,36),(1,37),(1,38),(1,39),(1,40),(1,41),(1,42),(1,43),(2,13),(2,17),(2,22),(2,23),(2,25),(2,31),(2,37),(2,103),(3,12),(3,16),(3,20),(3,21),(3,24),(3,30),(3,36),(3,103),(4,15),(4,19),(4,21),(4,23),(4,27),(4,33),(4,39),(4,102),(5,14),(5,18),(5,20),(5,22),(5,26),(5,32),(5,38),(5,102),(6,16),(6,17),(6,18),(6,19),(6,29),(6,35),(6,41),(6,101),(7,12),(7,13),(7,14),(7,15),(7,28),(7,34),(7,40),(7,101),(8,30),(8,31),(8,32),(8,33),(8,34),(8,35),(8,43),(8,100),(9,24),(9,25),(9,26),(9,27),(9,28),(9,29),(9,42),(9,100),(10,11),(10,100),(10,101),(10,102),(10,103),(11,104),(11,105),(11,106),(11,107),(12,44),(12,68),(12,80),(12,92),(12,93),(12,173),(13,45),(13,69),(13,81),(13,94),(13,95),(13,173),(14,46),(14,70),(14,82),(14,92),(14,94),(14,174),(15,47),(15,71),(15,83),(15,93),(15,95),(15,174),(16,48),(16,72),(16,84),(16,96),(16,97),(16,173),(17,49),(17,73),(17,85),(17,98),(17,99),(17,173),(18,50),(18,74),(18,86),(18,96),(18,98),(18,174),(19,51),(19,75),(19,87),(19,97),(19,99),(19,174),(20,52),(20,76),(20,88),(20,92),(20,96),(20,172),(21,53),(21,77),(21,89),(21,93),(21,97),(21,172),(22,54),(22,78),(22,90),(22,94),(22,98),(22,172),(23,55),(23,79),(23,91),(23,95),(23,99),(23,172),(24,56),(24,68),(24,72),(24,76),(24,77),(24,175),(25,57),(25,69),(25,73),(25,78),(25,79),(25,175),(26,58),(26,70),(26,74),(26,76),(26,78),(26,176),(27,59),(27,71),(27,75),(27,77),(27,79),(27,176),(28,60),(28,68),(28,69),(28,70),(28,71),(28,177),(29,61),(29,72),(29,73),(29,74),(29,75),(29,177),(30,62),(30,80),(30,84),(30,88),(30,89),(30,175),(31,63),(31,81),(31,85),(31,90),(31,91),(31,175),(32,64),(32,82),(32,86),(32,88),(32,90),(32,176),(33,65),(33,83),(33,87),(33,89),(33,91),(33,176),(34,66),(34,80),(34,81),(34,82),(34,83),(34,177),(35,67),(35,84),(35,85),(35,86),(35,87),(35,177),(36,44),(36,48),(36,52),(36,53),(36,56),(36,62),(36,104),(37,45),(37,49),(37,54),(37,55),(37,57),(37,63),(37,104),(38,46),(38,50),(38,52),(38,54),(38,58),(38,64),(38,105),(39,47),(39,51),(39,53),(39,55),(39,59),(39,65),(39,105),(40,44),(40,45),(40,46),(40,47),(40,60),(40,66),(40,106),(41,48),(41,49),(41,50),(41,51),(41,61),(41,67),(41,106),(42,56),(42,57),(42,58),(42,59),(42,60),(42,61),(42,107),(43,62),(43,63),(43,64),(43,65),(43,66),(43,67),(43,107),(44,124),(44,136),(44,148),(44,149),(44,179),(45,125),(45,137),(45,150),(45,151),(45,179),(46,126),(46,138),(46,148),(46,150),(46,180),(47,127),(47,139),(47,149),(47,151),(47,180),(48,128),(48,140),(48,152),(48,153),(48,179),(49,129),(49,141),(49,154),(49,155),(49,179),(50,130),(50,142),(50,152),(50,154),(50,180),(51,131),(51,143),(51,153),(51,155),(51,180),(52,132),(52,144),(52,148),(52,152),(52,178),(53,133),(53,145),(53,149),(53,153),(53,178),(54,134),(54,146),(54,150),(54,154),(54,178),(55,135),(55,147),(55,151),(55,155),(55,178),(56,124),(56,128),(56,132),(56,133),(56,181),(57,125),(57,129),(57,134),(57,135),(57,181),(58,126),(58,130),(58,132),(58,134),(58,182),(59,127),(59,131),(59,133),(59,135),(59,182),(60,124),(60,125),(60,126),(60,127),(60,183),(61,128),(61,129),(61,130),(61,131),(61,183),(62,136),(62,140),(62,144),(62,145),(62,181),(63,137),(63,141),(63,146),(63,147),(63,181),(64,138),(64,142),(64,144),(64,146),(64,182),(65,139),(65,143),(65,145),(65,147),(65,182),(66,136),(66,137),(66,138),(66,139),(66,183),(67,140),(67,141),(67,142),(67,143),(67,183),(68,108),(68,109),(68,124),(68,185),(69,110),(69,111),(69,125),(69,185),(70,108),(70,110),(70,126),(70,186),(71,109),(71,111),(71,127),(71,186),(72,112),(72,113),(72,128),(72,185),(73,114),(73,115),(73,129),(73,185),(74,112),(74,114),(74,130),(74,186),(75,113),(75,115),(75,131),(75,186),(76,108),(76,112),(76,132),(76,187),(77,109),(77,113),(77,133),(77,187),(78,110),(78,114),(78,134),(78,187),(79,111),(79,115),(79,135),(79,187),(80,116),(80,117),(80,136),(80,185),(81,118),(81,119),(81,137),(81,185),(82,116),(82,118),(82,138),(82,186),(83,117),(83,119),(83,139),(83,186),(84,120),(84,121),(84,140),(84,185),(85,122),(85,123),(85,141),(85,185),(86,120),(86,122),(86,142),(86,186),(87,121),(87,123),(87,143),(87,186),(88,116),(88,120),(88,144),(88,187),(89,117),(89,121),(89,145),(89,187),(90,118),(90,122),(90,146),(90,187),(91,119),(91,123),(91,147),(91,187),(92,108),(92,116),(92,148),(92,184),(93,109),(93,117),(93,149),(93,184),(94,110),(94,118),(94,150),(94,184),(95,111),(95,119),(95,151),(95,184),(96,112),(96,120),(96,152),(96,184),(97,113),(97,121),(97,153),(97,184),(98,114),(98,122),(98,154),(98,184),(99,115),(99,123),(99,155),(99,184),(100,107),(100,175),(100,176),(100,177),(101,106),(101,173),(101,174),(101,177),(102,105),(102,172),(102,174),(102,176),(103,104),(103,172),(103,173),(103,175),(104,178),(104,179),(104,181),(105,178),(105,180),(105,182),(106,179),(106,180),(106,183),(107,181),(107,182),(107,183),(108,156),(108,192),(109,157),(109,192),(110,158),(110,192),(111,159),(111,192),(112,160),(112,192),(113,161),(113,192),(114,162),(114,192),(115,163),(115,192),(116,164),(116,192),(117,165),(117,192),(118,166),(118,192),(119,167),(119,192),(120,168),(120,192),(121,169),(121,192),(122,170),(122,192),(123,171),(123,192),(124,156),(124,157),(124,189),(125,158),(125,159),(125,189),(126,156),(126,158),(126,190),(127,157),(127,159),(127,190),(128,160),(128,161),(128,189),(129,162),(129,163),(129,189),(130,160),(130,162),(130,190),(131,161),(131,163),(131,190),(132,156),(132,160),(132,191),(133,157),(133,161),(133,191),(134,158),(134,162),(134,191),(135,159),(135,163),(135,191),(136,164),(136,165),(136,189),(137,166),(137,167),(137,189),(138,164),(138,166),(138,190),(139,165),(139,167),(139,190),(140,168),(140,169),(140,189),(141,170),(141,171),(141,189),(142,168),(142,170),(142,190),(143,169),(143,171),(143,190),(144,164),(144,168),(144,191),(145,165),(145,169),(145,191),(146,166),(146,170),(146,191),(147,167),(147,171),(147,191),(148,156),(148,164),(148,188),(149,157),(149,165),(149,188),(150,158),(150,166),(150,188),(151,159),(151,167),(151,188),(152,160),(152,168),(152,188),(153,161),(153,169),(153,188),(154,162),(154,170),(154,188),(155,163),(155,171),(155,188),(156,193),(157,193),(158,193),(159,193),(160,193),(161,193),(162,193),(163,193),(164,193),(165,193),(166,193),(167,193),(168,193),(169,193),(170,193),(171,193),(172,178),(172,184),(172,187),(173,179),(173,184),(173,185),(174,180),(174,184),(174,186),(175,181),(175,185),(175,187),(176,182),(176,186),(176,187),(177,183),(177,185),(177,186),(178,188),(178,191),(179,188),(179,189),(180,188),(180,190),(181,189),(181,191),(182,190),(182,191),(183,189),(183,190),(184,188),(184,192),(185,189),(185,192),(186,190),(186,192),(187,191),(187,192),(188,193),(189,193),(190,193),(191,193),(192,193)],194)
 => ? = 0
[1,1,1,3,1] => [4,1,2] => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,11),(1,15),(1,20),(1,21),(1,23),(1,29),(1,69),(2,10),(2,14),(2,18),(2,19),(2,22),(2,28),(2,69),(3,13),(3,17),(3,19),(3,21),(3,25),(3,31),(3,68),(4,12),(4,16),(4,18),(4,20),(4,24),(4,30),(4,68),(5,14),(5,15),(5,16),(5,17),(5,27),(5,33),(5,67),(6,10),(6,11),(6,12),(6,13),(6,26),(6,32),(6,67),(7,28),(7,29),(7,30),(7,31),(7,32),(7,33),(7,66),(8,22),(8,23),(8,24),(8,25),(8,26),(8,27),(8,66),(9,66),(9,67),(9,68),(9,69),(10,34),(10,46),(10,58),(10,59),(10,87),(11,35),(11,47),(11,60),(11,61),(11,87),(12,36),(12,48),(12,58),(12,60),(12,88),(13,37),(13,49),(13,59),(13,61),(13,88),(14,38),(14,50),(14,62),(14,63),(14,87),(15,39),(15,51),(15,64),(15,65),(15,87),(16,40),(16,52),(16,62),(16,64),(16,88),(17,41),(17,53),(17,63),(17,65),(17,88),(18,42),(18,54),(18,58),(18,62),(18,86),(19,43),(19,55),(19,59),(19,63),(19,86),(20,44),(20,56),(20,60),(20,64),(20,86),(21,45),(21,57),(21,61),(21,65),(21,86),(22,34),(22,38),(22,42),(22,43),(22,89),(23,35),(23,39),(23,44),(23,45),(23,89),(24,36),(24,40),(24,42),(24,44),(24,90),(25,37),(25,41),(25,43),(25,45),(25,90),(26,34),(26,35),(26,36),(26,37),(26,91),(27,38),(27,39),(27,40),(27,41),(27,91),(28,46),(28,50),(28,54),(28,55),(28,89),(29,47),(29,51),(29,56),(29,57),(29,89),(30,48),(30,52),(30,54),(30,56),(30,90),(31,49),(31,53),(31,55),(31,57),(31,90),(32,46),(32,47),(32,48),(32,49),(32,91),(33,50),(33,51),(33,52),(33,53),(33,91),(34,70),(34,71),(34,93),(35,72),(35,73),(35,93),(36,70),(36,72),(36,94),(37,71),(37,73),(37,94),(38,74),(38,75),(38,93),(39,76),(39,77),(39,93),(40,74),(40,76),(40,94),(41,75),(41,77),(41,94),(42,70),(42,74),(42,95),(43,71),(43,75),(43,95),(44,72),(44,76),(44,95),(45,73),(45,77),(45,95),(46,78),(46,79),(46,93),(47,80),(47,81),(47,93),(48,78),(48,80),(48,94),(49,79),(49,81),(49,94),(50,82),(50,83),(50,93),(51,84),(51,85),(51,93),(52,82),(52,84),(52,94),(53,83),(53,85),(53,94),(54,78),(54,82),(54,95),(55,79),(55,83),(55,95),(56,80),(56,84),(56,95),(57,81),(57,85),(57,95),(58,70),(58,78),(58,92),(59,71),(59,79),(59,92),(60,72),(60,80),(60,92),(61,73),(61,81),(61,92),(62,74),(62,82),(62,92),(63,75),(63,83),(63,92),(64,76),(64,84),(64,92),(65,77),(65,85),(65,92),(66,89),(66,90),(66,91),(67,87),(67,88),(67,91),(68,86),(68,88),(68,90),(69,86),(69,87),(69,89),(70,96),(71,96),(72,96),(73,96),(74,96),(75,96),(76,96),(77,96),(78,96),(79,96),(80,96),(81,96),(82,96),(83,96),(84,96),(85,96),(86,92),(86,95),(87,92),(87,93),(88,92),(88,94),(89,93),(89,95),(90,94),(90,95),(91,93),(91,94),(92,96),(93,96),(94,96),(95,96)],97)
 => ? = 1
[1,1,2,1,1,1] => [3,4] => ([(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
 => 0
[1,1,2,1,2] => [3,3,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,11),(1,30),(1,31),(1,32),(1,33),(1,34),(1,35),(1,36),(2,10),(2,24),(2,25),(2,26),(2,27),(2,28),(2,29),(2,36),(3,13),(3,17),(3,22),(3,23),(3,25),(3,31),(3,78),(4,12),(4,16),(4,20),(4,21),(4,24),(4,30),(4,78),(5,15),(5,19),(5,21),(5,23),(5,27),(5,33),(5,77),(6,14),(6,18),(6,20),(6,22),(6,26),(6,32),(6,77),(7,16),(7,17),(7,18),(7,19),(7,29),(7,35),(7,76),(8,12),(8,13),(8,14),(8,15),(8,28),(8,34),(8,76),(9,10),(9,11),(9,76),(9,77),(9,78),(10,37),(10,79),(10,80),(10,81),(11,37),(11,82),(11,83),(11,84),(12,44),(12,56),(12,68),(12,69),(12,125),(13,45),(13,57),(13,70),(13,71),(13,125),(14,46),(14,58),(14,68),(14,70),(14,126),(15,47),(15,59),(15,69),(15,71),(15,126),(16,48),(16,60),(16,72),(16,73),(16,125),(17,49),(17,61),(17,74),(17,75),(17,125),(18,50),(18,62),(18,72),(18,74),(18,126),(19,51),(19,63),(19,73),(19,75),(19,126),(20,52),(20,64),(20,68),(20,72),(20,124),(21,53),(21,65),(21,69),(21,73),(21,124),(22,54),(22,66),(22,70),(22,74),(22,124),(23,55),(23,67),(23,71),(23,75),(23,124),(24,38),(24,44),(24,48),(24,52),(24,53),(24,79),(25,39),(25,45),(25,49),(25,54),(25,55),(25,79),(26,40),(26,46),(26,50),(26,52),(26,54),(26,80),(27,41),(27,47),(27,51),(27,53),(27,55),(27,80),(28,42),(28,44),(28,45),(28,46),(28,47),(28,81),(29,43),(29,48),(29,49),(29,50),(29,51),(29,81),(30,38),(30,56),(30,60),(30,64),(30,65),(30,82),(31,39),(31,57),(31,61),(31,66),(31,67),(31,82),(32,40),(32,58),(32,62),(32,64),(32,66),(32,83),(33,41),(33,59),(33,63),(33,65),(33,67),(33,83),(34,42),(34,56),(34,57),(34,58),(34,59),(34,84),(35,43),(35,60),(35,61),(35,62),(35,63),(35,84),(36,37),(36,38),(36,39),(36,40),(36,41),(36,42),(36,43),(37,113),(37,114),(37,115),(38,101),(38,105),(38,109),(38,110),(38,113),(39,102),(39,106),(39,111),(39,112),(39,113),(40,103),(40,107),(40,109),(40,111),(40,114),(41,104),(41,108),(41,110),(41,112),(41,114),(42,101),(42,102),(42,103),(42,104),(42,115),(43,105),(43,106),(43,107),(43,108),(43,115),(44,85),(44,86),(44,101),(44,128),(45,87),(45,88),(45,102),(45,128),(46,85),(46,87),(46,103),(46,129),(47,86),(47,88),(47,104),(47,129),(48,89),(48,90),(48,105),(48,128),(49,91),(49,92),(49,106),(49,128),(50,89),(50,91),(50,107),(50,129),(51,90),(51,92),(51,108),(51,129),(52,85),(52,89),(52,109),(52,127),(53,86),(53,90),(53,110),(53,127),(54,87),(54,91),(54,111),(54,127),(55,88),(55,92),(55,112),(55,127),(56,93),(56,94),(56,101),(56,131),(57,95),(57,96),(57,102),(57,131),(58,93),(58,95),(58,103),(58,132),(59,94),(59,96),(59,104),(59,132),(60,97),(60,98),(60,105),(60,131),(61,99),(61,100),(61,106),(61,131),(62,97),(62,99),(62,107),(62,132),(63,98),(63,100),(63,108),(63,132),(64,93),(64,97),(64,109),(64,130),(65,94),(65,98),(65,110),(65,130),(66,95),(66,99),(66,111),(66,130),(67,96),(67,100),(67,112),(67,130),(68,85),(68,93),(68,136),(69,86),(69,94),(69,136),(70,87),(70,95),(70,136),(71,88),(71,96),(71,136),(72,89),(72,97),(72,136),(73,90),(73,98),(73,136),(74,91),(74,99),(74,136),(75,92),(75,100),(75,136),(76,81),(76,84),(76,125),(76,126),(77,80),(77,83),(77,124),(77,126),(78,79),(78,82),(78,124),(78,125),(79,113),(79,127),(79,128),(80,114),(80,127),(80,129),(81,115),(81,128),(81,129),(82,113),(82,130),(82,131),(83,114),(83,130),(83,132),(84,115),(84,131),(84,132),(85,116),(85,137),(86,117),(86,137),(87,118),(87,137),(88,119),(88,137),(89,120),(89,137),(90,121),(90,137),(91,122),(91,137),(92,123),(92,137),(93,116),(93,138),(94,117),(94,138),(95,118),(95,138),(96,119),(96,138),(97,120),(97,138),(98,121),(98,138),(99,122),(99,138),(100,123),(100,138),(101,116),(101,117),(101,134),(102,118),(102,119),(102,134),(103,116),(103,118),(103,135),(104,117),(104,119),(104,135),(105,120),(105,121),(105,134),(106,122),(106,123),(106,134),(107,120),(107,122),(107,135),(108,121),(108,123),(108,135),(109,116),(109,120),(109,133),(110,117),(110,121),(110,133),(111,118),(111,122),(111,133),(112,119),(112,123),(112,133),(113,133),(113,134),(114,133),(114,135),(115,134),(115,135),(116,139),(117,139),(118,139),(119,139),(120,139),(121,139),(122,139),(123,139),(124,127),(124,130),(124,136),(125,128),(125,131),(125,136),(126,129),(126,132),(126,136),(127,133),(127,137),(128,134),(128,137),(129,135),(129,137),(130,133),(130,138),(131,134),(131,138),(132,135),(132,138),(133,139),(134,139),(135,139),(136,137),(136,138),(137,139),(138,139)],140)
 => ? = 0
[1,1,2,2,1] => [3,2,2] => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,9),(1,22),(1,23),(1,24),(1,25),(1,26),(1,27),(2,11),(2,15),(2,20),(2,21),(2,23),(2,50),(3,10),(3,14),(3,18),(3,19),(3,22),(3,50),(4,13),(4,17),(4,19),(4,21),(4,25),(4,49),(5,12),(5,16),(5,18),(5,20),(5,24),(5,49),(6,14),(6,15),(6,16),(6,17),(6,27),(6,48),(7,10),(7,11),(7,12),(7,13),(7,26),(7,48),(8,9),(8,48),(8,49),(8,50),(9,59),(9,60),(9,61),(10,28),(10,40),(10,41),(10,63),(11,29),(11,42),(11,43),(11,63),(12,30),(12,40),(12,42),(12,64),(13,31),(13,41),(13,43),(13,64),(14,32),(14,44),(14,45),(14,63),(15,33),(15,46),(15,47),(15,63),(16,34),(16,44),(16,46),(16,64),(17,35),(17,45),(17,47),(17,64),(18,36),(18,40),(18,44),(18,62),(19,37),(19,41),(19,45),(19,62),(20,38),(20,42),(20,46),(20,62),(21,39),(21,43),(21,47),(21,62),(22,28),(22,32),(22,36),(22,37),(22,59),(23,29),(23,33),(23,38),(23,39),(23,59),(24,30),(24,34),(24,36),(24,38),(24,60),(25,31),(25,35),(25,37),(25,39),(25,60),(26,28),(26,29),(26,30),(26,31),(26,61),(27,32),(27,33),(27,34),(27,35),(27,61),(28,51),(28,52),(28,66),(29,53),(29,54),(29,66),(30,51),(30,53),(30,67),(31,52),(31,54),(31,67),(32,55),(32,56),(32,66),(33,57),(33,58),(33,66),(34,55),(34,57),(34,67),(35,56),(35,58),(35,67),(36,51),(36,55),(36,65),(37,52),(37,56),(37,65),(38,53),(38,57),(38,65),(39,54),(39,58),(39,65),(40,51),(40,68),(41,52),(41,68),(42,53),(42,68),(43,54),(43,68),(44,55),(44,68),(45,56),(45,68),(46,57),(46,68),(47,58),(47,68),(48,61),(48,63),(48,64),(49,60),(49,62),(49,64),(50,59),(50,62),(50,63),(51,69),(52,69),(53,69),(54,69),(55,69),(56,69),(57,69),(58,69),(59,65),(59,66),(60,65),(60,67),(61,66),(61,67),(62,65),(62,68),(63,66),(63,68),(64,67),(64,68),(65,69),(66,69),(67,69),(68,69)],70)
 => ? = 1
[1,1,2,3] => [3,2,1,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 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 => ? = 0
[1,1,3,1,1] => [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
 => ? = 1
[1,1,3,2] => [3,1,2,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(1,14),(1,15),(1,16),(1,53),(1,54),(1,55),(1,56),(1,57),(1,58),(1,59),(1,60),(1,61),(2,19),(2,30),(2,31),(2,36),(2,37),(2,45),(2,46),(2,55),(2,140),(2,143),(3,18),(3,27),(3,29),(3,33),(3,35),(3,44),(3,46),(3,54),(3,139),(3,142),(4,17),(4,26),(4,28),(4,32),(4,34),(4,44),(4,45),(4,53),(4,138),(4,141),(5,22),(5,28),(5,29),(5,38),(5,39),(5,47),(5,48),(5,56),(5,135),(5,143),(6,21),(6,26),(6,30),(6,40),(6,42),(6,47),(6,49),(6,57),(6,136),(6,142),(7,20),(7,27),(7,31),(7,41),(7,43),(7,48),(7,49),(7,58),(7,137),(7,141),(8,25),(8,34),(8,35),(8,40),(8,41),(8,50),(8,51),(8,59),(8,135),(8,140),(9,24),(9,32),(9,36),(9,38),(9,43),(9,50),(9,52),(9,60),(9,136),(9,139),(10,23),(10,33),(10,37),(10,39),(10,42),(10,51),(10,52),(10,61),(10,137),(10,138),(11,16),(11,23),(11,24),(11,25),(11,62),(11,141),(11,142),(11,143),(12,15),(12,20),(12,21),(12,22),(12,62),(12,138),(12,139),(12,140),(13,14),(13,17),(13,18),(13,19),(13,62),(13,135),(13,136),(13,137),(14,75),(14,76),(14,77),(14,144),(14,184),(14,185),(14,186),(15,78),(15,79),(15,80),(15,144),(15,187),(15,188),(15,189),(16,81),(16,82),(16,83),(16,144),(16,190),(16,191),(16,192),(17,63),(17,64),(17,75),(17,193),(17,195),(17,259),(18,63),(18,65),(18,76),(18,194),(18,196),(18,260),(19,64),(19,65),(19,77),(19,197),(19,198),(19,261),(20,66),(20,68),(20,78),(20,202),(20,203),(20,259),(21,67),(21,68),(21,79),(21,201),(21,204),(21,260),(22,66),(22,67),(22,80),(22,199),(22,200),(22,261),(23,69),(23,71),(23,81),(23,208),(23,209),(23,259),(24,70),(24,71),(24,82),(24,207),(24,210),(24,260),(25,69),(25,70),(25,83),(25,205),(25,206),(25,261),(26,93),(26,112),(26,114),(26,129),(26,193),(26,201),(26,256),(27,94),(27,113),(27,115),(27,130),(27,194),(27,202),(27,256),(28,95),(28,111),(28,114),(28,131),(28,195),(28,199),(28,257),(29,96),(29,111),(29,115),(29,132),(29,196),(29,200),(29,258),(30,97),(30,112),(30,116),(30,133),(30,197),(30,204),(30,258),(31,98),(31,113),(31,116),(31,134),(31,198),(31,203),(31,257),(32,99),(32,118),(32,123),(32,131),(32,193),(32,207),(32,253),(33,100),(33,119),(33,124),(33,132),(33,194),(33,208),(33,253),(34,101),(34,117),(34,123),(34,129),(34,195),(34,205),(34,254),(35,102),(35,117),(35,124),(35,130),(35,196),(35,206),(35,255),(36,103),(36,118),(36,125),(36,134),(36,197),(36,210),(36,255),(37,104),(37,119),(37,125),(37,133),(37,198),(37,209),(37,254),(38,105),(38,121),(38,128),(38,131),(38,200),(38,210),(38,251),(39,106),(39,120),(39,128),(39,132),(39,199),(39,209),(39,250),(40,107),(40,122),(40,126),(40,129),(40,204),(40,206),(40,251),(41,108),(41,122),(41,127),(41,130),(41,203),(41,205),(41,250),(42,109),(42,120),(42,126),(42,133),(42,201),(42,208),(42,252),(43,110),(43,121),(43,127),(43,134),(43,202),(43,207),(43,252),(44,63),(44,72),(44,86),(44,111),(44,117),(44,253),(44,256),(45,64),(45,72),(45,84),(45,112),(45,118),(45,254),(45,257),(46,65),(46,72),(46,85),(46,113),(46,119),(46,255),(46,258),(47,67),(47,73),(47,87),(47,114),(47,120),(47,251),(47,258),(48,66),(48,73),(48,88),(48,115),(48,121),(48,250),(48,257),(49,68),(49,73),(49,89),(49,116),(49,122),(49,252),(49,256),(50,70),(50,74),(50,90),(50,123),(50,127),(50,251),(50,255),(51,69),(51,74),(51,91),(51,124),(51,126),(51,250),(51,254),(52,71),(52,74),(52,92),(52,125),(52,128),(52,252),(52,253),(53,75),(53,84),(53,86),(53,93),(53,95),(53,99),(53,101),(53,187),(53,190),(54,76),(54,85),(54,86),(54,94),(54,96),(54,100),(54,102),(54,188),(54,191),(55,77),(55,84),(55,85),(55,97),(55,98),(55,103),(55,104),(55,189),(55,192),(56,80),(56,87),(56,88),(56,95),(56,96),(56,105),(56,106),(56,184),(56,192),(57,79),(57,87),(57,89),(57,93),(57,97),(57,107),(57,109),(57,185),(57,191),(58,78),(58,88),(58,89),(58,94),(58,98),(58,108),(58,110),(58,186),(58,190),(59,83),(59,90),(59,91),(59,101),(59,102),(59,107),(59,108),(59,184),(59,189),(60,82),(60,90),(60,92),(60,99),(60,103),(60,105),(60,110),(60,185),(60,188),(61,81),(61,91),(61,92),(61,100),(61,104),(61,106),(61,109),(61,186),(61,187),(62,144),(62,259),(62,260),(62,261),(63,145),(63,159),(63,213),(63,300),(64,145),(64,157),(64,211),(64,298),(65,145),(65,158),(65,212),(65,299),(66,146),(66,160),(66,215),(66,298),(67,146),(67,161),(67,214),(67,299),(68,146),(68,162),(68,216),(68,300),(69,147),(69,163),(69,218),(69,298),(70,147),(70,164),(70,217),(70,299),(71,147),(71,165),(71,219),(71,300),(72,145),(72,148),(72,293),(72,294),(73,146),(73,149),(73,292),(73,294),(74,147),(74,150),(74,292),(74,293),(75,157),(75,159),(75,220),(75,222),(75,280),(76,158),(76,159),(76,221),(76,223),(76,281),(77,157),(77,158),(77,224),(77,225),(77,282),(78,160),(78,162),(78,229),(78,230),(78,280),(79,161),(79,162),(79,228),(79,231),(79,281),(80,160),(80,161),(80,226),(80,227),(80,282),(81,163),(81,165),(81,235),(81,236),(81,280),(82,164),(82,165),(82,234),(82,237),(82,281),(83,163),(83,164),(83,232),(83,233),(83,282),(84,148),(84,157),(84,167),(84,173),(84,275),(84,278),(85,148),(85,158),(85,168),(85,174),(85,276),(85,279),(86,148),(86,159),(86,166),(86,172),(86,274),(86,277),(87,149),(87,161),(87,169),(87,175),(87,272),(87,279),(88,149),(88,160),(88,170),(88,176),(88,271),(88,278),(89,149),(89,162),(89,171),(89,177),(89,273),(89,277),(90,150),(90,164),(90,178),(90,182),(90,272),(90,276),(91,150),(91,163),(91,179),(91,181),(91,271),(91,275),(92,150),(92,165),(92,180),(92,183),(92,273),(92,274),(93,151),(93,167),(93,169),(93,220),(93,228),(93,277),(94,152),(94,168),(94,170),(94,221),(94,229),(94,277),(95,153),(95,166),(95,169),(95,222),(95,226),(95,278),(96,154),(96,166),(96,170),(96,223),(96,227),(96,279),(97,155),(97,167),(97,171),(97,224),(97,231),(97,279),(98,156),(98,168),(98,171),(98,225),(98,230),(98,278),(99,153),(99,173),(99,178),(99,220),(99,234),(99,274),(100,154),(100,174),(100,179),(100,221),(100,235),(100,274),(101,151),(101,172),(101,178),(101,222),(101,232),(101,275),(102,152),(102,172),(102,179),(102,223),(102,233),(102,276),(103,156),(103,173),(103,180),(103,224),(103,237),(103,276),(104,155),(104,174),(104,180),(104,225),(104,236),(104,275),(105,153),(105,176),(105,183),(105,227),(105,237),(105,272),(106,154),(106,175),(106,183),(106,226),(106,236),(106,271),(107,151),(107,177),(107,181),(107,231),(107,233),(107,272),(108,152),(108,177),(108,182),(108,230),(108,232),(108,271),(109,155),(109,175),(109,181),(109,228),(109,235),(109,273),(110,156),(110,176),(110,182),(110,229),(110,234),(110,273),(111,166),(111,213),(111,267),(111,294),(112,167),(112,211),(112,265),(112,294),(113,168),(113,212),(113,266),(113,294),(114,169),(114,214),(114,262),(114,294),(115,170),(115,215),(115,263),(115,294),(116,171),(116,216),(116,264),(116,294),(117,172),(117,213),(117,268),(117,293),(118,173),(118,211),(118,269),(118,293),(119,174),(119,212),(119,270),(119,293),(120,175),(120,214),(120,270),(120,292),(121,176),(121,215),(121,269),(121,292),(122,177),(122,216),(122,268),(122,292),(123,178),(123,217),(123,262),(123,293),(124,179),(124,218),(124,263),(124,293),(125,180),(125,219),(125,264),(125,293),(126,181),(126,218),(126,265),(126,292),(127,182),(127,217),(127,266),(127,292),(128,183),(128,219),(128,267),(128,292),(129,151),(129,262),(129,265),(129,268),(130,152),(130,263),(130,266),(130,268),(131,153),(131,262),(131,267),(131,269),(132,154),(132,263),(132,267),(132,270),(133,155),(133,264),(133,265),(133,270),(134,156),(134,264),(134,266),(134,269),(135,184),(135,195),(135,196),(135,250),(135,251),(135,261),(136,185),(136,193),(136,197),(136,251),(136,252),(136,260),(137,186),(137,194),(137,198),(137,250),(137,252),(137,259),(138,187),(138,199),(138,201),(138,253),(138,254),(138,259),(139,188),(139,200),(139,202),(139,253),(139,255),(139,260),(140,189),(140,203),(140,204),(140,254),(140,255),(140,261),(141,190),(141,205),(141,207),(141,256),(141,257),(141,259),(142,191),(142,206),(142,208),(142,256),(142,258),(142,260),(143,192),(143,209),(143,210),(143,257),(143,258),(143,261),(144,280),(144,281),(144,282),(145,238),(145,304),(146,239),(146,304),(147,240),(147,304),(148,238),(148,296),(148,297),(149,239),(149,295),(149,297),(150,240),(150,295),(150,296),(151,283),(151,287),(151,289),(152,284),(152,288),(152,289),(153,283),(153,286),(153,290),(154,284),(154,286),(154,291),(155,285),(155,287),(155,291),(156,285),(156,288),(156,290),(157,238),(157,241),(157,301),(158,238),(158,242),(158,302),(159,238),(159,243),(159,303),(160,239),(160,245),(160,301),(161,239),(161,244),(161,302),(162,239),(162,246),(162,303),(163,240),(163,248),(163,301),(164,240),(164,247),(164,302),(165,240),(165,249),(165,303),(166,243),(166,286),(166,297),(167,241),(167,287),(167,297),(168,242),(168,288),(168,297),(169,244),(169,283),(169,297),(170,245),(170,284),(170,297),(171,246),(171,285),(171,297),(172,243),(172,289),(172,296),(173,241),(173,290),(173,296),(174,242),(174,291),(174,296),(175,244),(175,291),(175,295),(176,245),(176,290),(176,295),(177,246),(177,289),(177,295),(178,247),(178,283),(178,296),(179,248),(179,284),(179,296),(180,249),(180,285),(180,296),(181,248),(181,287),(181,295),(182,247),(182,288),(182,295),(183,249),(183,286),(183,295),(184,222),(184,223),(184,271),(184,272),(184,282),(185,220),(185,224),(185,272),(185,273),(185,281),(186,221),(186,225),(186,271),(186,273),(186,280),(187,226),(187,228),(187,274),(187,275),(187,280),(188,227),(188,229),(188,274),(188,276),(188,281),(189,230),(189,231),(189,275),(189,276),(189,282),(190,232),(190,234),(190,277),(190,278),(190,280),(191,233),(191,235),(191,277),(191,279),(191,281),(192,236),(192,237),(192,278),(192,279),(192,282),(193,211),(193,220),(193,262),(193,300),(194,212),(194,221),(194,263),(194,300),(195,213),(195,222),(195,262),(195,298),(196,213),(196,223),(196,263),(196,299),(197,211),(197,224),(197,264),(197,299),(198,212),(198,225),(198,264),(198,298),(199,214),(199,226),(199,267),(199,298),(200,215),(200,227),(200,267),(200,299),(201,214),(201,228),(201,265),(201,300),(202,215),(202,229),(202,266),(202,300),(203,216),(203,230),(203,266),(203,298),(204,216),(204,231),(204,265),(204,299),(205,217),(205,232),(205,268),(205,298),(206,218),(206,233),(206,268),(206,299),(207,217),(207,234),(207,269),(207,300),(208,218),(208,235),(208,270),(208,300),(209,219),(209,236),(209,270),(209,298),(210,219),(210,237),(210,269),(210,299),(211,241),(211,304),(212,242),(212,304),(213,243),(213,304),(214,244),(214,304),(215,245),(215,304),(216,246),(216,304),(217,247),(217,304),(218,248),(218,304),(219,249),(219,304),(220,241),(220,283),(220,303),(221,242),(221,284),(221,303),(222,243),(222,283),(222,301),(223,243),(223,284),(223,302),(224,241),(224,285),(224,302),(225,242),(225,285),(225,301),(226,244),(226,286),(226,301),(227,245),(227,286),(227,302),(228,244),(228,287),(228,303),(229,245),(229,288),(229,303),(230,246),(230,288),(230,301),(231,246),(231,287),(231,302),(232,247),(232,289),(232,301),(233,248),(233,289),(233,302),(234,247),(234,290),(234,303),(235,248),(235,291),(235,303),(236,249),(236,291),(236,301),(237,249),(237,290),(237,302),(238,305),(239,305),(240,305),(241,305),(242,305),(243,305),(244,305),(245,305),(246,305),(247,305),(248,305),(249,305),(250,263),(250,271),(250,292),(250,298),(251,262),(251,272),(251,292),(251,299),(252,264),(252,273),(252,292),(252,300),(253,267),(253,274),(253,293),(253,300),(254,265),(254,275),(254,293),(254,298),(255,266),(255,276),(255,293),(255,299),(256,268),(256,277),(256,294),(256,300),(257,269),(257,278),(257,294),(257,298),(258,270),(258,279),(258,294),(258,299),(259,280),(259,298),(259,300),(260,281),(260,299),(260,300),(261,282),(261,298),(261,299),(262,283),(262,304),(263,284),(263,304),(264,285),(264,304),(265,287),(265,304),(266,288),(266,304),(267,286),(267,304),(268,289),(268,304),(269,290),(269,304),(270,291),(270,304),(271,284),(271,295),(271,301),(272,283),(272,295),(272,302),(273,285),(273,295),(273,303),(274,286),(274,296),(274,303),(275,287),(275,296),(275,301),(276,288),(276,296),(276,302),(277,289),(277,297),(277,303),(278,290),(278,297),(278,301),(279,291),(279,297),(279,302),(280,301),(280,303),(281,302),(281,303),(282,301),(282,302),(283,305),(284,305),(285,305),(286,305),(287,305),(288,305),(289,305),(290,305),(291,305),(292,295),(292,304),(293,296),(293,304),(294,297),(294,304),(295,305),(296,305),(297,305),(298,301),(298,304),(299,302),(299,304),(300,303),(300,304),(301,305),(302,305),(303,305),(304,305)],306)
 => ? = 1
[1,1,4,1] => [3,1,1,2] => ([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,15),(1,26),(1,27),(1,32),(1,33),(1,41),(1,42),(1,91),(1,94),(2,14),(2,23),(2,25),(2,29),(2,31),(2,40),(2,42),(2,90),(2,93),(3,13),(3,22),(3,24),(3,28),(3,30),(3,40),(3,41),(3,89),(3,92),(4,18),(4,24),(4,25),(4,34),(4,35),(4,43),(4,44),(4,86),(4,94),(5,17),(5,22),(5,26),(5,36),(5,38),(5,43),(5,45),(5,87),(5,93),(6,16),(6,23),(6,27),(6,37),(6,39),(6,44),(6,45),(6,88),(6,92),(7,21),(7,30),(7,31),(7,36),(7,37),(7,46),(7,47),(7,86),(7,91),(8,20),(8,28),(8,32),(8,34),(8,39),(8,46),(8,48),(8,87),(8,90),(9,19),(9,29),(9,33),(9,35),(9,38),(9,47),(9,48),(9,88),(9,89),(10,19),(10,20),(10,21),(10,49),(10,92),(10,93),(10,94),(11,16),(11,17),(11,18),(11,49),(11,89),(11,90),(11,91),(12,13),(12,14),(12,15),(12,49),(12,86),(12,87),(12,88),(13,53),(13,54),(13,98),(13,100),(13,134),(14,53),(14,55),(14,99),(14,101),(14,135),(15,54),(15,55),(15,102),(15,103),(15,136),(16,56),(16,58),(16,107),(16,108),(16,134),(17,57),(17,58),(17,106),(17,109),(17,135),(18,56),(18,57),(18,104),(18,105),(18,136),(19,59),(19,61),(19,113),(19,114),(19,134),(20,60),(20,61),(20,112),(20,115),(20,135),(21,59),(21,60),(21,110),(21,111),(21,136),(22,63),(22,65),(22,80),(22,98),(22,106),(22,131),(23,64),(23,66),(23,81),(23,99),(23,107),(23,131),(24,62),(24,65),(24,82),(24,100),(24,104),(24,132),(25,62),(25,66),(25,83),(25,101),(25,105),(25,133),(26,63),(26,67),(26,84),(26,102),(26,109),(26,133),(27,64),(27,67),(27,85),(27,103),(27,108),(27,132),(28,69),(28,74),(28,82),(28,98),(28,112),(28,128),(29,70),(29,75),(29,83),(29,99),(29,113),(29,128),(30,68),(30,74),(30,80),(30,100),(30,110),(30,129),(31,68),(31,75),(31,81),(31,101),(31,111),(31,130),(32,69),(32,76),(32,85),(32,102),(32,115),(32,130),(33,70),(33,76),(33,84),(33,103),(33,114),(33,129),(34,72),(34,79),(34,82),(34,105),(34,115),(34,126),(35,71),(35,79),(35,83),(35,104),(35,114),(35,125),(36,73),(36,77),(36,80),(36,109),(36,111),(36,126),(37,73),(37,78),(37,81),(37,108),(37,110),(37,125),(38,71),(38,77),(38,84),(38,106),(38,113),(38,127),(39,72),(39,78),(39,85),(39,107),(39,112),(39,127),(40,50),(40,53),(40,62),(40,68),(40,128),(40,131),(41,50),(41,54),(41,63),(41,69),(41,129),(41,132),(42,50),(42,55),(42,64),(42,70),(42,130),(42,133),(43,51),(43,57),(43,65),(43,71),(43,126),(43,133),(44,51),(44,56),(44,66),(44,72),(44,125),(44,132),(45,51),(45,58),(45,67),(45,73),(45,127),(45,131),(46,52),(46,60),(46,74),(46,78),(46,126),(46,130),(47,52),(47,59),(47,75),(47,77),(47,125),(47,129),(48,52),(48,61),(48,76),(48,79),(48,127),(48,128),(49,134),(49,135),(49,136),(50,95),(50,147),(50,148),(51,96),(51,146),(51,148),(52,97),(52,146),(52,147),(53,95),(53,118),(53,151),(54,95),(54,116),(54,149),(55,95),(55,117),(55,150),(56,96),(56,120),(56,149),(57,96),(57,119),(57,150),(58,96),(58,121),(58,151),(59,97),(59,123),(59,149),(60,97),(60,122),(60,150),(61,97),(61,124),(61,151),(62,118),(62,142),(62,148),(63,116),(63,140),(63,148),(64,117),(64,141),(64,148),(65,119),(65,137),(65,148),(66,120),(66,138),(66,148),(67,121),(67,139),(67,148),(68,118),(68,143),(68,147),(69,116),(69,144),(69,147),(70,117),(70,145),(70,147),(71,119),(71,145),(71,146),(72,120),(72,144),(72,146),(73,121),(73,143),(73,146),(74,122),(74,137),(74,147),(75,123),(75,138),(75,147),(76,124),(76,139),(76,147),(77,123),(77,140),(77,146),(78,122),(78,141),(78,146),(79,124),(79,142),(79,146),(80,137),(80,140),(80,143),(81,138),(81,141),(81,143),(82,137),(82,142),(82,144),(83,138),(83,142),(83,145),(84,139),(84,140),(84,145),(85,139),(85,141),(85,144),(86,100),(86,101),(86,125),(86,126),(86,136),(87,98),(87,102),(87,126),(87,127),(87,135),(88,99),(88,103),(88,125),(88,127),(88,134),(89,104),(89,106),(89,128),(89,129),(89,134),(90,105),(90,107),(90,128),(90,130),(90,135),(91,108),(91,109),(91,129),(91,130),(91,136),(92,110),(92,112),(92,131),(92,132),(92,134),(93,111),(93,113),(93,131),(93,133),(93,135),(94,114),(94,115),(94,132),(94,133),(94,136),(95,152),(96,152),(97,152),(98,116),(98,137),(98,151),(99,117),(99,138),(99,151),(100,118),(100,137),(100,149),(101,118),(101,138),(101,150),(102,116),(102,139),(102,150),(103,117),(103,139),(103,149),(104,119),(104,142),(104,149),(105,120),(105,142),(105,150),(106,119),(106,140),(106,151),(107,120),(107,141),(107,151),(108,121),(108,141),(108,149),(109,121),(109,140),(109,150),(110,122),(110,143),(110,149),(111,123),(111,143),(111,150),(112,122),(112,144),(112,151),(113,123),(113,145),(113,151),(114,124),(114,145),(114,149),(115,124),(115,144),(115,150),(116,152),(117,152),(118,152),(119,152),(120,152),(121,152),(122,152),(123,152),(124,152),(125,138),(125,146),(125,149),(126,137),(126,146),(126,150),(127,139),(127,146),(127,151),(128,142),(128,147),(128,151),(129,140),(129,147),(129,149),(130,141),(130,147),(130,150),(131,143),(131,148),(131,151),(132,144),(132,148),(132,149),(133,145),(133,148),(133,150),(134,149),(134,151),(135,150),(135,151),(136,149),(136,150),(137,152),(138,152),(139,152),(140,152),(141,152),(142,152),(143,152),(144,152),(145,152),(146,152),(147,152),(148,152),(149,152),(150,152),(151,152)],153)
 => ? = 2
[1,2,1,1,1,1] => [2,5] => ([(4,6),(5,6)],7)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 0
[1,2,1,1,2] => [2,4,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,11),(1,18),(1,21),(1,24),(1,27),(1,29),(1,30),(2,10),(2,17),(2,20),(2,23),(2,26),(2,28),(2,30),(3,9),(3,16),(3,19),(3,22),(3,25),(3,28),(3,29),(4,14),(4,15),(4,19),(4,20),(4,21),(4,60),(5,12),(5,13),(5,16),(5,17),(5,18),(5,60),(6,13),(6,15),(6,25),(6,26),(6,27),(6,59),(7,12),(7,14),(7,22),(7,23),(7,24),(7,59),(8,9),(8,10),(8,11),(8,59),(8,60),(9,31),(9,32),(9,78),(9,81),(10,31),(10,33),(10,79),(10,82),(11,32),(11,33),(11,80),(11,83),(12,35),(12,36),(12,37),(12,94),(13,38),(13,39),(13,40),(13,94),(14,41),(14,42),(14,43),(14,94),(15,44),(15,45),(15,46),(15,94),(16,35),(16,38),(16,47),(16,48),(16,78),(17,36),(17,39),(17,47),(17,49),(17,79),(18,37),(18,40),(18,48),(18,49),(18,80),(19,41),(19,44),(19,50),(19,51),(19,78),(20,42),(20,45),(20,50),(20,52),(20,79),(21,43),(21,46),(21,51),(21,52),(21,80),(22,35),(22,41),(22,53),(22,54),(22,81),(23,36),(23,42),(23,53),(23,55),(23,82),(24,37),(24,43),(24,54),(24,55),(24,83),(25,38),(25,44),(25,56),(25,57),(25,81),(26,39),(26,45),(26,56),(26,58),(26,82),(27,40),(27,46),(27,57),(27,58),(27,83),(28,31),(28,34),(28,47),(28,50),(28,53),(28,56),(29,32),(29,34),(29,48),(29,51),(29,54),(29,57),(30,33),(30,34),(30,49),(30,52),(30,55),(30,58),(31,77),(31,86),(31,89),(32,77),(32,84),(32,87),(33,77),(33,85),(33,88),(34,73),(34,74),(34,75),(34,76),(34,77),(35,61),(35,62),(35,97),(36,61),(36,63),(36,98),(37,62),(37,63),(37,99),(38,64),(38,65),(38,97),(39,64),(39,66),(39,98),(40,65),(40,66),(40,99),(41,67),(41,68),(41,97),(42,67),(42,69),(42,98),(43,68),(43,69),(43,99),(44,70),(44,71),(44,97),(45,70),(45,72),(45,98),(46,71),(46,72),(46,99),(47,61),(47,64),(47,73),(47,86),(48,62),(48,65),(48,73),(48,84),(49,63),(49,66),(49,73),(49,85),(50,67),(50,70),(50,74),(50,86),(51,68),(51,71),(51,74),(51,84),(52,69),(52,72),(52,74),(52,85),(53,61),(53,67),(53,75),(53,89),(54,62),(54,68),(54,75),(54,87),(55,63),(55,69),(55,75),(55,88),(56,64),(56,70),(56,76),(56,89),(57,65),(57,71),(57,76),(57,87),(58,66),(58,72),(58,76),(58,88),(59,81),(59,82),(59,83),(59,94),(60,78),(60,79),(60,80),(60,94),(61,90),(61,102),(62,90),(62,100),(63,90),(63,101),(64,91),(64,102),(65,91),(65,100),(66,91),(66,101),(67,92),(67,102),(68,92),(68,100),(69,92),(69,101),(70,93),(70,102),(71,93),(71,100),(72,93),(72,101),(73,90),(73,91),(73,95),(74,92),(74,93),(74,95),(75,90),(75,92),(75,96),(76,91),(76,93),(76,96),(77,95),(77,96),(78,84),(78,86),(78,97),(79,85),(79,86),(79,98),(80,84),(80,85),(80,99),(81,87),(81,89),(81,97),(82,88),(82,89),(82,98),(83,87),(83,88),(83,99),(84,95),(84,100),(85,95),(85,101),(86,95),(86,102),(87,96),(87,100),(88,96),(88,101),(89,96),(89,102),(90,103),(91,103),(92,103),(93,103),(94,97),(94,98),(94,99),(95,103),(96,103),(97,100),(97,102),(98,101),(98,102),(99,100),(99,101),(100,103),(101,103),(102,103)],104)
 => ? = 0
[1,2,1,2,1] => [2,3,2] => ([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(1,9),(1,36),(1,37),(2,12),(2,16),(2,21),(2,22),(2,37),(3,11),(3,15),(3,19),(3,20),(3,37),(4,14),(4,18),(4,20),(4,22),(4,36),(5,13),(5,17),(5,19),(5,21),(5,36),(6,9),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,10),(7,11),(7,12),(7,13),(7,14),(8,35),(8,38),(8,39),(9,35),(9,40),(9,41),(10,31),(10,32),(10,33),(10,34),(10,35),(11,23),(11,24),(11,31),(11,38),(12,25),(12,26),(12,32),(12,38),(13,23),(13,25),(13,33),(13,39),(14,24),(14,26),(14,34),(14,39),(15,27),(15,28),(15,31),(15,40),(16,29),(16,30),(16,32),(16,40),(17,27),(17,29),(17,33),(17,41),(18,28),(18,30),(18,34),(18,41),(19,23),(19,27),(19,48),(20,24),(20,28),(20,48),(21,25),(21,29),(21,48),(22,26),(22,30),(22,48),(23,42),(23,49),(24,43),(24,49),(25,44),(25,49),(26,45),(26,49),(27,42),(27,50),(28,43),(28,50),(29,44),(29,50),(30,45),(30,50),(31,42),(31,43),(31,46),(32,44),(32,45),(32,46),(33,42),(33,44),(33,47),(34,43),(34,45),(34,47),(35,46),(35,47),(36,39),(36,41),(36,48),(37,38),(37,40),(37,48),(38,46),(38,49),(39,47),(39,49),(40,46),(40,50),(41,47),(41,50),(42,51),(43,51),(44,51),(45,51),(46,51),(47,51),(48,49),(48,50),(49,51),(50,51)],52)
 => ? = 1
[1,2,1,3] => [2,3,1,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 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 => ? = 0
[1,2,2,1,1] => [2,2,3] => ([(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
 => ? = 1
[1,2,2,2] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(1,13),(1,26),(1,27),(1,47),(1,48),(1,49),(1,50),(1,51),(1,52),(1,53),(1,54),(2,18),(2,19),(2,34),(2,35),(2,43),(2,44),(2,45),(2,46),(2,54),(2,56),(3,16),(3,17),(3,32),(3,33),(3,39),(3,40),(3,41),(3,42),(3,53),(3,56),(4,21),(4,23),(4,31),(4,37),(4,40),(4,44),(4,48),(4,120),(4,121),(5,20),(5,22),(5,30),(5,36),(5,39),(5,43),(5,47),(5,119),(5,121),(6,20),(6,24),(6,28),(6,37),(6,41),(6,45),(6,49),(6,117),(6,122),(7,21),(7,25),(7,29),(7,36),(7,42),(7,46),(7,50),(7,118),(7,122),(8,15),(8,24),(8,25),(8,33),(8,35),(8,38),(8,52),(8,119),(8,120),(9,14),(9,22),(9,23),(9,32),(9,34),(9,38),(9,51),(9,117),(9,118),(10,16),(10,18),(10,26),(10,29),(10,30),(10,55),(10,117),(10,120),(11,17),(11,19),(11,27),(11,28),(11,31),(11,55),(11,118),(11,119),(12,13),(12,14),(12,15),(12,55),(12,56),(12,121),(12,122),(13,62),(13,63),(13,135),(13,136),(13,137),(13,161),(14,57),(14,62),(14,129),(14,178),(14,236),(15,57),(15,63),(15,130),(15,179),(15,237),(16,72),(16,98),(16,99),(16,170),(16,173),(16,211),(17,73),(17,97),(17,100),(17,171),(17,172),(17,211),(18,74),(18,102),(18,103),(18,174),(18,177),(18,211),(19,75),(19,101),(19,104),(19,175),(19,176),(19,211),(20,58),(20,60),(20,76),(20,166),(20,168),(20,212),(21,59),(21,61),(21,77),(21,167),(21,169),(21,212),(22,68),(22,89),(22,93),(22,129),(22,166),(22,209),(23,69),(23,90),(23,94),(23,129),(23,167),(23,210),(24,70),(24,91),(24,95),(24,130),(24,168),(24,210),(25,71),(25,92),(25,96),(25,130),(25,169),(25,209),(26,72),(26,74),(26,106),(26,107),(26,135),(26,162),(26,165),(27,73),(27,75),(27,105),(27,108),(27,135),(27,163),(27,164),(28,97),(28,101),(28,105),(28,116),(28,168),(28,236),(29,98),(29,102),(29,106),(29,115),(29,169),(29,236),(30,99),(30,103),(30,107),(30,115),(30,166),(30,237),(31,100),(31,104),(31,108),(31,116),(31,167),(31,237),(32,89),(32,90),(32,109),(32,113),(32,170),(32,171),(32,178),(33,91),(33,92),(33,110),(33,113),(33,172),(33,173),(33,179),(34,93),(34,94),(34,111),(34,114),(34,174),(34,175),(34,178),(35,95),(35,96),(35,112),(35,114),(35,176),(35,177),(35,179),(36,64),(36,66),(36,79),(36,115),(36,209),(36,212),(37,65),(37,67),(37,80),(37,116),(37,210),(37,212),(38,57),(38,78),(38,113),(38,114),(38,209),(38,210),(39,58),(39,64),(39,81),(39,89),(39,99),(39,172),(39,213),(40,59),(40,65),(40,82),(40,90),(40,100),(40,173),(40,213),(41,58),(41,65),(41,83),(41,91),(41,97),(41,170),(41,214),(42,59),(42,64),(42,84),(42,92),(42,98),(42,171),(42,214),(43,60),(43,66),(43,85),(43,93),(43,103),(43,176),(43,213),(44,61),(44,67),(44,86),(44,94),(44,104),(44,177),(44,213),(45,60),(45,67),(45,87),(45,95),(45,101),(45,174),(45,214),(46,61),(46,66),(46,88),(46,96),(46,102),(46,175),(46,214),(47,68),(47,76),(47,79),(47,81),(47,85),(47,107),(47,136),(47,164),(48,69),(48,77),(48,80),(48,82),(48,86),(48,108),(48,136),(48,165),(49,70),(49,76),(49,80),(49,83),(49,87),(49,105),(49,137),(49,162),(50,71),(50,77),(50,79),(50,84),(50,88),(50,106),(50,137),(50,163),(51,62),(51,68),(51,69),(51,78),(51,109),(51,111),(51,162),(51,163),(52,63),(52,70),(52,71),(52,78),(52,110),(52,112),(52,164),(52,165),(53,72),(53,73),(53,81),(53,82),(53,83),(53,84),(53,109),(53,110),(53,161),(54,74),(54,75),(54,85),(54,86),(54,87),(54,88),(54,111),(54,112),(54,161),(55,135),(55,211),(55,236),(55,237),(56,161),(56,178),(56,179),(56,211),(56,213),(56,214),(57,138),(57,180),(57,252),(58,157),(58,181),(58,183),(58,246),(59,158),(59,182),(59,184),(59,246),(60,159),(60,185),(60,187),(60,246),(61,160),(61,186),(61,188),(61,246),(62,138),(62,189),(62,203),(62,238),(63,138),(63,190),(63,204),(63,239),(64,123),(64,131),(64,224),(64,246),(65,124),(65,132),(65,225),(65,246),(66,125),(66,133),(66,226),(66,246),(67,126),(67,134),(67,227),(67,246),(68,147),(68,151),(68,199),(68,203),(68,234),(69,148),(69,152),(69,200),(69,203),(69,235),(70,149),(70,153),(70,201),(70,204),(70,235),(71,150),(71,154),(71,202),(71,204),(71,234),(72,140),(72,141),(72,191),(72,194),(72,233),(73,139),(73,142),(73,192),(73,193),(73,233),(74,144),(74,145),(74,195),(74,198),(74,233),(75,143),(75,146),(75,196),(75,197),(75,233),(76,157),(76,159),(76,199),(76,201),(76,230),(77,158),(77,160),(77,200),(77,202),(77,230),(78,138),(78,155),(78,156),(78,234),(78,235),(79,127),(79,131),(79,133),(79,230),(79,234),(80,128),(80,132),(80,134),(80,230),(80,235),(81,131),(81,141),(81,147),(81,157),(81,193),(81,231),(82,132),(82,142),(82,148),(82,158),(82,194),(82,231),(83,132),(83,139),(83,149),(83,157),(83,191),(83,232),(84,131),(84,140),(84,150),(84,158),(84,192),(84,232),(85,133),(85,145),(85,151),(85,159),(85,197),(85,231),(86,134),(86,146),(86,152),(86,160),(86,198),(86,231),(87,134),(87,143),(87,153),(87,159),(87,195),(87,232),(88,133),(88,144),(88,154),(88,160),(88,196),(88,232),(89,147),(89,181),(89,224),(89,228),(90,148),(90,182),(90,225),(90,228),(91,149),(91,183),(91,225),(91,229),(92,150),(92,184),(92,224),(92,229),(93,151),(93,185),(93,226),(93,228),(94,152),(94,186),(94,227),(94,228),(95,153),(95,187),(95,227),(95,229),(96,154),(96,188),(96,226),(96,229),(97,124),(97,139),(97,183),(97,247),(98,123),(98,140),(98,184),(98,247),(99,123),(99,141),(99,181),(99,248),(100,124),(100,142),(100,182),(100,248),(101,126),(101,143),(101,187),(101,247),(102,125),(102,144),(102,188),(102,247),(103,125),(103,145),(103,185),(103,248),(104,126),(104,146),(104,186),(104,248),(105,128),(105,139),(105,143),(105,201),(105,238),(106,127),(106,140),(106,144),(106,202),(106,238),(107,127),(107,141),(107,145),(107,199),(107,239),(108,128),(108,142),(108,146),(108,200),(108,239),(109,147),(109,148),(109,155),(109,189),(109,191),(109,192),(110,149),(110,150),(110,155),(110,190),(110,193),(110,194),(111,151),(111,152),(111,156),(111,189),(111,195),(111,196),(112,153),(112,154),(112,156),(112,190),(112,197),(112,198),(113,155),(113,180),(113,224),(113,225),(114,156),(114,180),(114,226),(114,227),(115,123),(115,125),(115,127),(115,252),(116,124),(116,126),(116,128),(116,252),(117,162),(117,166),(117,170),(117,174),(117,210),(117,236),(118,163),(118,167),(118,171),(118,175),(118,209),(118,236),(119,164),(119,168),(119,172),(119,176),(119,209),(119,237),(120,165),(120,169),(120,173),(120,177),(120,210),(120,237),(121,129),(121,136),(121,212),(121,213),(121,237),(122,130),(122,137),(122,212),(122,214),(122,236),(123,205),(123,254),(124,206),(124,254),(125,207),(125,254),(126,208),(126,254),(127,205),(127,207),(127,253),(128,206),(128,208),(128,253),(129,203),(129,228),(129,252),(130,204),(130,229),(130,252),(131,205),(131,242),(131,249),(132,206),(132,243),(132,249),(133,207),(133,244),(133,249),(134,208),(134,245),(134,249),(135,233),(135,238),(135,239),(136,203),(136,230),(136,231),(136,239),(137,204),(137,230),(137,232),(137,238),(138,215),(138,253),(139,206),(139,218),(139,250),(140,205),(140,219),(140,250),(141,205),(141,216),(141,251),(142,206),(142,217),(142,251),(143,208),(143,222),(143,250),(144,207),(144,223),(144,250),(145,207),(145,220),(145,251),(146,208),(146,221),(146,251),(147,216),(147,240),(147,242),(148,217),(148,240),(148,243),(149,218),(149,241),(149,243),(150,219),(150,241),(150,242),(151,220),(151,240),(151,244),(152,221),(152,240),(152,245),(153,222),(153,241),(153,245),(154,223),(154,241),(154,244),(155,215),(155,242),(155,243),(156,215),(156,244),(156,245),(157,216),(157,218),(157,249),(158,217),(158,219),(158,249),(159,220),(159,222),(159,249),(160,221),(160,223),(160,249),(161,189),(161,190),(161,231),(161,232),(161,233),(162,191),(162,195),(162,199),(162,235),(162,238),(163,192),(163,196),(163,200),(163,234),(163,238),(164,193),(164,197),(164,201),(164,234),(164,239),(165,194),(165,198),(165,202),(165,235),(165,239),(166,181),(166,185),(166,199),(166,252),(167,182),(167,186),(167,200),(167,252),(168,183),(168,187),(168,201),(168,252),(169,184),(169,188),(169,202),(169,252),(170,181),(170,191),(170,225),(170,247),(171,182),(171,192),(171,224),(171,247),(172,183),(172,193),(172,224),(172,248),(173,184),(173,194),(173,225),(173,248),(174,185),(174,195),(174,227),(174,247),(175,186),(175,196),(175,226),(175,247),(176,187),(176,197),(176,226),(176,248),(177,188),(177,198),(177,227),(177,248),(178,180),(178,189),(178,228),(178,247),(179,180),(179,190),(179,229),(179,248),(180,215),(180,254),(181,216),(181,254),(182,217),(182,254),(183,218),(183,254),(184,219),(184,254),(185,220),(185,254),(186,221),(186,254),(187,222),(187,254),(188,223),(188,254),(189,215),(189,240),(189,250),(190,215),(190,241),(190,251),(191,216),(191,243),(191,250),(192,217),(192,242),(192,250),(193,218),(193,242),(193,251),(194,219),(194,243),(194,251),(195,220),(195,245),(195,250),(196,221),(196,244),(196,250),(197,222),(197,244),(197,251),(198,223),(198,245),(198,251),(199,216),(199,220),(199,253),(200,217),(200,221),(200,253),(201,218),(201,222),(201,253),(202,219),(202,223),(202,253),(203,240),(203,253),(204,241),(204,253),(205,255),(206,255),(207,255),(208,255),(209,224),(209,226),(209,234),(209,252),(210,225),(210,227),(210,235),(210,252),(211,233),(211,247),(211,248),(212,230),(212,246),(212,252),(213,228),(213,231),(213,246),(213,248),(214,229),(214,232),(214,246),(214,247),(215,255),(216,255),(217,255),(218,255),(219,255),(220,255),(221,255),(222,255),(223,255),(224,242),(224,254),(225,243),(225,254),(226,244),(226,254),(227,245),(227,254),(228,240),(228,254),(229,241),(229,254),(230,249),(230,253),(231,240),(231,249),(231,251),(232,241),(232,249),(232,250),(233,250),(233,251),(234,242),(234,244),(234,253),(235,243),(235,245),(235,253),(236,238),(236,247),(236,252),(237,239),(237,248),(237,252),(238,250),(238,253),(239,251),(239,253),(240,255),(241,255),(242,255),(243,255),(244,255),(245,255),(246,249),(246,254),(247,250),(247,254),(248,251),(248,254),(249,255),(250,255),(251,255),(252,253),(252,254),(253,255),(254,255)],256)
 => ? = 1
[1,2,3,1] => [2,2,1,2] => ([(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,16),(1,17),(1,30),(1,31),(1,39),(1,40),(1,41),(1,42),(1,49),(2,14),(2,15),(2,28),(2,29),(2,35),(2,36),(2,37),(2,38),(2,49),(3,19),(3,23),(3,27),(3,34),(3,36),(3,40),(3,74),(3,79),(4,18),(4,22),(4,26),(4,33),(4,35),(4,39),(4,74),(4,78),(5,21),(5,22),(5,24),(5,34),(5,37),(5,41),(5,75),(5,76),(6,20),(6,23),(6,25),(6,33),(6,38),(6,42),(6,75),(6,77),(7,13),(7,20),(7,21),(7,29),(7,31),(7,32),(7,78),(7,79),(8,12),(8,18),(8,19),(8,28),(8,30),(8,32),(8,76),(8,77),(9,14),(9,16),(9,25),(9,26),(9,48),(9,76),(9,79),(10,15),(10,17),(10,24),(10,27),(10,48),(10,77),(10,78),(11,12),(11,13),(11,48),(11,49),(11,74),(11,75),(12,43),(12,84),(12,98),(12,121),(13,43),(13,85),(13,99),(13,122),(14,55),(14,56),(14,86),(14,89),(14,109),(15,54),(15,57),(15,87),(15,88),(15,109),(16,59),(16,60),(16,90),(16,93),(16,109),(17,58),(17,61),(17,91),(17,92),(17,109),(18,62),(18,66),(18,84),(18,94),(18,113),(19,63),(19,67),(19,84),(19,95),(19,114),(20,64),(20,68),(20,85),(20,97),(20,113),(21,65),(21,69),(21,85),(21,96),(21,114),(22,50),(22,52),(22,94),(22,96),(22,110),(23,51),(23,53),(23,95),(23,97),(23,110),(24,54),(24,58),(24,71),(24,96),(24,121),(25,55),(25,59),(25,70),(25,97),(25,121),(26,56),(26,60),(26,70),(26,94),(26,122),(27,57),(27,61),(27,71),(27,95),(27,122),(28,62),(28,63),(28,72),(28,86),(28,87),(28,98),(29,64),(29,65),(29,72),(29,88),(29,89),(29,99),(30,66),(30,67),(30,73),(30,90),(30,91),(30,98),(31,68),(31,69),(31,73),(31,92),(31,93),(31,99),(32,43),(32,72),(32,73),(32,113),(32,114),(33,44),(33,46),(33,70),(33,110),(33,113),(34,45),(34,47),(34,71),(34,110),(34,114),(35,44),(35,50),(35,56),(35,62),(35,88),(35,111),(36,45),(36,51),(36,57),(36,63),(36,89),(36,111),(37,45),(37,50),(37,54),(37,65),(37,86),(37,112),(38,44),(38,51),(38,55),(38,64),(38,87),(38,112),(39,46),(39,52),(39,60),(39,66),(39,92),(39,111),(40,47),(40,53),(40,61),(40,67),(40,93),(40,111),(41,47),(41,52),(41,58),(41,69),(41,90),(41,112),(42,46),(42,53),(42,59),(42,68),(42,91),(42,112),(43,100),(43,126),(44,80),(44,117),(44,123),(45,81),(45,118),(45,123),(46,82),(46,119),(46,123),(47,83),(47,120),(47,123),(48,109),(48,121),(48,122),(49,98),(49,99),(49,109),(49,111),(49,112),(50,101),(50,103),(50,123),(51,102),(51,104),(51,123),(52,105),(52,107),(52,123),(53,106),(53,108),(53,123),(54,81),(54,103),(54,124),(55,80),(55,104),(55,124),(56,80),(56,101),(56,125),(57,81),(57,102),(57,125),(58,83),(58,107),(58,124),(59,82),(59,108),(59,124),(60,82),(60,105),(60,125),(61,83),(61,106),(61,125),(62,101),(62,115),(62,117),(63,102),(63,115),(63,118),(64,104),(64,116),(64,117),(65,103),(65,116),(65,118),(66,105),(66,115),(66,119),(67,106),(67,115),(67,120),(68,108),(68,116),(68,119),(69,107),(69,116),(69,120),(70,80),(70,82),(70,126),(71,81),(71,83),(71,126),(72,100),(72,117),(72,118),(73,100),(73,119),(73,120),(74,84),(74,110),(74,111),(74,122),(75,85),(75,110),(75,112),(75,121),(76,86),(76,90),(76,94),(76,114),(76,121),(77,87),(77,91),(77,95),(77,113),(77,121),(78,88),(78,92),(78,96),(78,113),(78,122),(79,89),(79,93),(79,97),(79,114),(79,122),(80,127),(81,127),(82,127),(83,127),(84,115),(84,126),(85,116),(85,126),(86,101),(86,118),(86,124),(87,102),(87,117),(87,124),(88,103),(88,117),(88,125),(89,104),(89,118),(89,125),(90,105),(90,120),(90,124),(91,106),(91,119),(91,124),(92,107),(92,119),(92,125),(93,108),(93,120),(93,125),(94,101),(94,105),(94,126),(95,102),(95,106),(95,126),(96,103),(96,107),(96,126),(97,104),(97,108),(97,126),(98,100),(98,115),(98,124),(99,100),(99,116),(99,125),(100,127),(101,127),(102,127),(103,127),(104,127),(105,127),(106,127),(107,127),(108,127),(109,124),(109,125),(110,123),(110,126),(111,115),(111,123),(111,125),(112,116),(112,123),(112,124),(113,117),(113,119),(113,126),(114,118),(114,120),(114,126),(115,127),(116,127),(117,127),(118,127),(119,127),(120,127),(121,124),(121,126),(122,125),(122,126),(123,127),(124,127),(125,127),(126,127)],128)
 => ? = 2
[1,2,4] => [2,2,1,1,1] => ([(0,4),(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 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 => ? = 0
[1,3,1,1,1] => [2,1,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
 => ? = 1
[1,3,1,2] => [2,1,3,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(1,21),(1,22),(1,23),(1,39),(1,40),(1,41),(1,42),(1,43),(1,44),(1,45),(2,18),(2,19),(2,20),(2,33),(2,34),(2,35),(2,36),(2,37),(2,38),(2,45),(3,16),(3,17),(3,26),(3,32),(3,35),(3,41),(3,90),(3,91),(4,13),(4,15),(4,25),(4,31),(4,34),(4,40),(4,89),(4,91),(5,12),(5,14),(5,24),(5,30),(5,33),(5,39),(5,89),(5,90),(6,14),(6,15),(6,29),(6,32),(6,36),(6,42),(6,92),(6,93),(7,12),(7,16),(7,27),(7,31),(7,37),(7,43),(7,92),(7,94),(8,13),(8,17),(8,28),(8,30),(8,38),(8,44),(8,93),(8,94),(9,18),(9,21),(9,24),(9,28),(9,46),(9,91),(9,92),(10,19),(10,22),(10,25),(10,27),(10,46),(10,90),(10,93),(11,20),(11,23),(11,26),(11,29),(11,46),(11,89),(11,94),(12,47),(12,53),(12,116),(12,120),(12,158),(13,48),(13,54),(13,117),(13,121),(13,158),(14,49),(14,55),(14,116),(14,119),(14,159),(15,50),(15,56),(15,117),(15,119),(15,160),(16,51),(16,57),(16,118),(16,120),(16,160),(17,52),(17,58),(17,118),(17,121),(17,159),(18,74),(18,78),(18,86),(18,124),(18,125),(18,134),(19,75),(19,77),(19,87),(19,123),(19,126),(19,134),(20,76),(20,79),(20,88),(20,122),(20,127),(20,134),(21,80),(21,84),(21,86),(21,130),(21,131),(21,135),(22,81),(22,83),(22,87),(22,129),(22,132),(22,135),(23,82),(23,85),(23,88),(23,128),(23,133),(23,135),(24,59),(24,74),(24,80),(24,116),(24,174),(25,60),(25,75),(25,81),(25,117),(25,174),(26,61),(26,76),(26,82),(26,118),(26,174),(27,60),(27,77),(27,83),(27,120),(27,173),(28,59),(28,78),(28,84),(28,121),(28,173),(29,61),(29,79),(29,85),(29,119),(29,173),(30,59),(30,62),(30,65),(30,158),(30,159),(31,60),(31,63),(31,66),(31,158),(31,160),(32,61),(32,64),(32,67),(32,159),(32,160),(33,47),(33,49),(33,62),(33,68),(33,74),(33,122),(33,123),(34,48),(34,50),(34,63),(34,69),(34,75),(34,122),(34,124),(35,51),(35,52),(35,64),(35,70),(35,76),(35,123),(35,124),(36,49),(36,50),(36,64),(36,71),(36,79),(36,125),(36,126),(37,47),(37,51),(37,63),(37,72),(37,77),(37,125),(37,127),(38,48),(38,52),(38,62),(38,73),(38,78),(38,126),(38,127),(39,53),(39,55),(39,65),(39,68),(39,80),(39,128),(39,129),(40,54),(40,56),(40,66),(40,69),(40,81),(40,128),(40,130),(41,57),(41,58),(41,67),(41,70),(41,82),(41,129),(41,130),(42,55),(42,56),(42,67),(42,71),(42,85),(42,131),(42,132),(43,53),(43,57),(43,66),(43,72),(43,83),(43,131),(43,133),(44,54),(44,58),(44,65),(44,73),(44,84),(44,132),(44,133),(45,68),(45,69),(45,70),(45,71),(45,72),(45,73),(45,86),(45,87),(45,88),(46,134),(46,135),(46,173),(46,174),(47,110),(47,143),(47,147),(47,167),(48,111),(48,144),(48,148),(48,167),(49,112),(49,143),(49,146),(49,168),(50,113),(50,144),(50,146),(50,169),(51,114),(51,145),(51,147),(51,169),(52,115),(52,145),(52,148),(52,168),(53,110),(53,149),(53,153),(53,170),(54,111),(54,150),(54,154),(54,170),(55,112),(55,149),(55,152),(55,171),(56,113),(56,150),(56,152),(56,172),(57,114),(57,151),(57,153),(57,172),(58,115),(58,151),(58,154),(58,171),(59,98),(59,101),(59,184),(60,99),(60,102),(60,184),(61,100),(61,103),(61,184),(62,95),(62,98),(62,167),(62,168),(63,96),(63,99),(63,167),(63,169),(64,97),(64,100),(64,168),(64,169),(65,95),(65,101),(65,170),(65,171),(66,96),(66,102),(66,170),(66,172),(67,97),(67,103),(67,171),(67,172),(68,95),(68,104),(68,110),(68,112),(68,137),(68,138),(69,96),(69,105),(69,111),(69,113),(69,137),(69,139),(70,97),(70,106),(70,114),(70,115),(70,138),(70,139),(71,97),(71,109),(71,112),(71,113),(71,140),(71,141),(72,96),(72,107),(72,110),(72,114),(72,140),(72,142),(73,95),(73,108),(73,111),(73,115),(73,141),(73,142),(74,98),(74,104),(74,143),(74,175),(75,99),(75,105),(75,144),(75,175),(76,100),(76,106),(76,145),(76,175),(77,99),(77,107),(77,147),(77,176),(78,98),(78,108),(78,148),(78,176),(79,100),(79,109),(79,146),(79,176),(80,101),(80,104),(80,149),(80,177),(81,102),(81,105),(81,150),(81,177),(82,103),(82,106),(82,151),(82,177),(83,102),(83,107),(83,153),(83,178),(84,101),(84,108),(84,154),(84,178),(85,103),(85,109),(85,152),(85,178),(86,104),(86,108),(86,136),(86,139),(86,140),(87,105),(87,107),(87,136),(87,138),(87,141),(88,106),(88,109),(88,136),(88,137),(88,142),(89,119),(89,122),(89,128),(89,158),(89,174),(90,120),(90,123),(90,129),(90,159),(90,174),(91,121),(91,124),(91,130),(91,160),(91,174),(92,116),(92,125),(92,131),(92,160),(92,173),(93,117),(93,126),(93,132),(93,159),(93,173),(94,118),(94,127),(94,133),(94,158),(94,173),(95,155),(95,179),(95,180),(96,156),(96,179),(96,181),(97,157),(97,180),(97,181),(98,155),(98,185),(99,156),(99,185),(100,157),(100,185),(101,155),(101,186),(102,156),(102,186),(103,157),(103,186),(104,155),(104,161),(104,182),(105,156),(105,162),(105,182),(106,157),(106,163),(106,182),(107,156),(107,165),(107,183),(108,155),(108,166),(108,183),(109,157),(109,164),(109,183),(110,161),(110,165),(110,179),(111,162),(111,166),(111,179),(112,161),(112,164),(112,180),(113,162),(113,164),(113,181),(114,163),(114,165),(114,181),(115,163),(115,166),(115,180),(116,143),(116,149),(116,184),(117,144),(117,150),(117,184),(118,145),(118,151),(118,184),(119,146),(119,152),(119,184),(120,147),(120,153),(120,184),(121,148),(121,154),(121,184),(122,137),(122,146),(122,167),(122,175),(123,138),(123,147),(123,168),(123,175),(124,139),(124,148),(124,169),(124,175),(125,140),(125,143),(125,169),(125,176),(126,141),(126,144),(126,168),(126,176),(127,142),(127,145),(127,167),(127,176),(128,137),(128,152),(128,170),(128,177),(129,138),(129,153),(129,171),(129,177),(130,139),(130,154),(130,172),(130,177),(131,140),(131,149),(131,172),(131,178),(132,141),(132,150),(132,171),(132,178),(133,142),(133,151),(133,170),(133,178),(134,136),(134,175),(134,176),(135,136),(135,177),(135,178),(136,182),(136,183),(137,164),(137,179),(137,182),(138,165),(138,180),(138,182),(139,166),(139,181),(139,182),(140,161),(140,181),(140,183),(141,162),(141,180),(141,183),(142,163),(142,179),(142,183),(143,161),(143,185),(144,162),(144,185),(145,163),(145,185),(146,164),(146,185),(147,165),(147,185),(148,166),(148,185),(149,161),(149,186),(150,162),(150,186),(151,163),(151,186),(152,164),(152,186),(153,165),(153,186),(154,166),(154,186),(155,187),(156,187),(157,187),(158,167),(158,170),(158,184),(159,168),(159,171),(159,184),(160,169),(160,172),(160,184),(161,187),(162,187),(163,187),(164,187),(165,187),(166,187),(167,179),(167,185),(168,180),(168,185),(169,181),(169,185),(170,179),(170,186),(171,180),(171,186),(172,181),(172,186),(173,176),(173,178),(173,184),(174,175),(174,177),(174,184),(175,182),(175,185),(176,183),(176,185),(177,182),(177,186),(178,183),(178,186),(179,187),(180,187),(181,187),(182,187),(183,187),(184,185),(184,186),(185,187),(186,187)],188)
 => ? = 1
[1,3,2,1] => [2,1,2,2] => ([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,17),(1,18),(1,19),(1,29),(1,30),(1,31),(1,32),(1,33),(1,34),(2,15),(2,16),(2,22),(2,28),(2,31),(2,55),(2,56),(3,12),(3,14),(3,21),(3,27),(3,30),(3,54),(3,56),(4,11),(4,13),(4,20),(4,26),(4,29),(4,54),(4,55),(5,13),(5,14),(5,25),(5,28),(5,32),(5,57),(5,58),(6,11),(6,15),(6,23),(6,27),(6,33),(6,57),(6,59),(7,12),(7,16),(7,24),(7,26),(7,34),(7,58),(7,59),(8,17),(8,20),(8,24),(8,41),(8,56),(8,57),(9,18),(9,21),(9,23),(9,41),(9,55),(9,58),(10,19),(10,22),(10,25),(10,41),(10,54),(10,59),(11,35),(11,70),(11,74),(11,82),(12,36),(12,71),(12,75),(12,82),(13,37),(13,70),(13,73),(13,83),(14,38),(14,71),(14,73),(14,84),(15,39),(15,72),(15,74),(15,84),(16,40),(16,72),(16,75),(16,83),(17,45),(17,49),(17,63),(17,66),(17,67),(18,46),(18,48),(18,63),(18,65),(18,68),(19,47),(19,50),(19,63),(19,64),(19,69),(20,45),(20,51),(20,70),(20,89),(21,46),(21,52),(21,71),(21,89),(22,47),(22,53),(22,72),(22,89),(23,48),(23,52),(23,74),(23,88),(24,49),(24,51),(24,75),(24,88),(25,50),(25,53),(25,73),(25,88),(26,42),(26,51),(26,82),(26,83),(27,43),(27,52),(27,82),(27,84),(28,44),(28,53),(28,83),(28,84),(29,35),(29,37),(29,42),(29,45),(29,64),(29,65),(30,36),(30,38),(30,43),(30,46),(30,64),(30,66),(31,39),(31,40),(31,44),(31,47),(31,65),(31,66),(32,37),(32,38),(32,44),(32,50),(32,67),(32,68),(33,35),(33,39),(33,43),(33,48),(33,67),(33,69),(34,36),(34,40),(34,42),(34,49),(34,68),(34,69),(35,76),(35,80),(35,85),(36,77),(36,81),(36,85),(37,76),(37,79),(37,86),(38,77),(38,79),(38,87),(39,78),(39,80),(39,87),(40,78),(40,81),(40,86),(41,63),(41,88),(41,89),(42,60),(42,85),(42,86),(43,61),(43,85),(43,87),(44,62),(44,86),(44,87),(45,60),(45,76),(45,90),(46,61),(46,77),(46,90),(47,62),(47,78),(47,90),(48,61),(48,80),(48,91),(49,60),(49,81),(49,91),(50,62),(50,79),(50,91),(51,60),(51,92),(52,61),(52,92),(53,62),(53,92),(54,64),(54,73),(54,82),(54,89),(55,65),(55,74),(55,83),(55,89),(56,66),(56,75),(56,84),(56,89),(57,67),(57,70),(57,84),(57,88),(58,68),(58,71),(58,83),(58,88),(59,69),(59,72),(59,82),(59,88),(60,93),(61,93),(62,93),(63,90),(63,91),(64,79),(64,85),(64,90),(65,80),(65,86),(65,90),(66,81),(66,87),(66,90),(67,76),(67,87),(67,91),(68,77),(68,86),(68,91),(69,78),(69,85),(69,91),(70,76),(70,92),(71,77),(71,92),(72,78),(72,92),(73,79),(73,92),(74,80),(74,92),(75,81),(75,92),(76,93),(77,93),(78,93),(79,93),(80,93),(81,93),(82,85),(82,92),(83,86),(83,92),(84,87),(84,92),(85,93),(86,93),(87,93),(88,91),(88,92),(89,90),(89,92),(90,93),(91,93),(92,93)],94)
 => ? = 2
[1,3,3] => [2,1,2,1,1] => ([(0,5),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 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 => ? = 1
[1,4,1,1] => [2,1,1,3] => ([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,12),(1,15),(1,28),(1,31),(1,34),(2,11),(2,14),(2,28),(2,30),(2,33),(3,10),(3,13),(3,28),(3,29),(3,32),(4,10),(4,16),(4,19),(4,21),(4,30),(4,31),(5,11),(5,17),(5,20),(5,22),(5,29),(5,31),(6,12),(6,18),(6,23),(6,24),(6,29),(6,30),(7,13),(7,16),(7,20),(7,23),(7,33),(7,34),(8,14),(8,17),(8,19),(8,24),(8,32),(8,34),(9,15),(9,18),(9,21),(9,22),(9,32),(9,33),(10,25),(10,35),(10,45),(11,26),(11,36),(11,45),(12,27),(12,37),(12,45),(13,25),(13,38),(13,44),(14,26),(14,39),(14,44),(15,27),(15,40),(15,44),(16,25),(16,42),(16,43),(17,26),(17,41),(17,43),(18,27),(18,41),(18,42),(19,35),(19,39),(19,43),(20,36),(20,38),(20,43),(21,35),(21,40),(21,42),(22,36),(22,40),(22,41),(23,37),(23,38),(23,42),(24,37),(24,39),(24,41),(25,46),(26,46),(27,46),(28,44),(28,45),(29,38),(29,41),(29,45),(30,39),(30,42),(30,45),(31,40),(31,43),(31,45),(32,35),(32,41),(32,44),(33,36),(33,42),(33,44),(34,37),(34,43),(34,44),(35,46),(36,46),(37,46),(38,46),(39,46),(40,46),(41,46),(42,46),(43,46),(44,46),(45,46)],47)
 => ? = 3
[1,4,2] => [2,1,1,2,1] => ([(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(1,21),(1,25),(1,33),(1,37),(1,54),(1,56),(1,140),(1,143),(1,147),(1,153),(2,20),(2,24),(2,32),(2,36),(2,53),(2,55),(2,140),(2,142),(2,146),(2,152),(3,23),(3,27),(3,35),(3,39),(3,53),(3,57),(3,141),(3,143),(3,144),(3,150),(4,22),(4,26),(4,34),(4,38),(4,54),(4,58),(4,141),(4,142),(4,145),(4,151),(5,30),(5,31),(5,42),(5,43),(5,52),(5,60),(5,142),(5,143),(5,149),(5,155),(6,28),(6,29),(6,40),(6,41),(6,52),(6,59),(6,140),(6,141),(6,148),(6,154),(7,17),(7,22),(7,23),(7,29),(7,45),(7,62),(7,67),(7,68),(7,146),(7,147),(7,149),(8,16),(8,20),(8,21),(8,28),(8,44),(8,61),(8,65),(8,66),(8,144),(8,145),(8,149),(9,18),(9,24),(9,26),(9,31),(9,46),(9,63),(9,69),(9,71),(9,144),(9,147),(9,148),(10,19),(10,25),(10,27),(10,30),(10,47),(10,64),(10,70),(10,72),(10,145),(10,146),(10,148),(11,16),(11,32),(11,33),(11,40),(11,48),(11,62),(11,69),(11,70),(11,150),(11,151),(11,155),(12,17),(12,34),(12,35),(12,41),(12,49),(12,61),(12,71),(12,72),(12,152),(12,153),(12,155),(13,18),(13,36),(13,38),(13,43),(13,50),(13,64),(13,65),(13,67),(13,150),(13,153),(13,154),(14,19),(14,37),(14,39),(14,42),(14,51),(14,63),(14,66),(14,68),(14,151),(14,152),(14,154),(15,44),(15,45),(15,46),(15,47),(15,48),(15,49),(15,50),(15,51),(15,55),(15,56),(15,57),(15,58),(15,59),(15,60),(16,88),(16,89),(16,96),(16,136),(16,292),(16,293),(16,297),(17,90),(17,91),(17,97),(17,137),(17,294),(17,295),(17,297),(18,92),(18,94),(18,99),(18,138),(18,292),(18,295),(18,296),(19,93),(19,95),(19,98),(19,139),(19,293),(19,294),(19,296),(20,76),(20,88),(20,112),(20,184),(20,208),(20,232),(20,325),(21,77),(21,89),(21,113),(21,185),(21,209),(21,232),(21,326),(22,78),(22,90),(22,114),(22,186),(22,211),(22,233),(22,325),(23,79),(23,91),(23,115),(23,187),(23,210),(23,233),(23,326),(24,80),(24,92),(24,116),(24,190),(24,208),(24,234),(24,324),(25,81),(25,93),(25,117),(25,191),(25,209),(25,235),(25,324),(26,82),(26,94),(26,118),(26,188),(26,211),(26,234),(26,323),(27,83),(27,95),(27,119),(27,189),(27,210),(27,235),(27,323),(28,84),(28,96),(28,120),(28,192),(28,212),(28,232),(28,323),(29,85),(29,97),(29,121),(29,193),(29,212),(29,233),(29,324),(30,86),(30,98),(30,122),(30,194),(30,213),(30,235),(30,325),(31,87),(31,99),(31,123),(31,195),(31,213),(31,234),(31,326),(32,80),(32,88),(32,124),(32,198),(32,214),(32,236),(32,329),(33,81),(33,89),(33,125),(33,199),(33,215),(33,236),(33,330),(34,82),(34,90),(34,126),(34,196),(34,217),(34,237),(34,329),(35,83),(35,91),(35,127),(35,197),(35,216),(35,237),(35,330),(36,76),(36,92),(36,128),(36,202),(36,214),(36,238),(36,328),(37,77),(37,93),(37,129),(37,203),(37,215),(37,239),(37,328),(38,78),(38,94),(38,130),(38,200),(38,217),(38,238),(38,327),(39,79),(39,95),(39,131),(39,201),(39,216),(39,239),(39,327),(40,85),(40,96),(40,132),(40,204),(40,218),(40,236),(40,327),(41,84),(41,97),(41,133),(41,205),(41,218),(41,237),(41,328),(42,87),(42,98),(42,134),(42,207),(42,219),(42,239),(42,329),(43,86),(43,99),(43,135),(43,206),(43,219),(43,238),(43,330),(44,100),(44,104),(44,105),(44,112),(44,113),(44,120),(44,136),(44,220),(44,221),(44,225),(45,101),(45,106),(45,107),(45,114),(45,115),(45,121),(45,137),(45,222),(45,223),(45,225),(46,102),(46,108),(46,110),(46,116),(46,118),(46,123),(46,138),(46,220),(46,223),(46,224),(47,103),(47,109),(47,111),(47,117),(47,119),(47,122),(47,139),(47,221),(47,222),(47,224),(48,101),(48,108),(48,109),(48,124),(48,125),(48,132),(48,136),(48,226),(48,227),(48,231),(49,100),(49,110),(49,111),(49,126),(49,127),(49,133),(49,137),(49,228),(49,229),(49,231),(50,103),(50,104),(50,106),(50,128),(50,130),(50,135),(50,138),(50,226),(50,229),(50,230),(51,102),(51,105),(51,107),(51,129),(51,131),(51,134),(51,139),(51,227),(51,228),(51,230),(52,75),(52,212),(52,213),(52,218),(52,219),(52,304),(53,73),(53,208),(53,210),(53,214),(53,216),(53,304),(54,74),(54,209),(54,211),(54,215),(54,217),(54,304),(55,73),(55,112),(55,116),(55,124),(55,128),(55,156),(55,158),(55,222),(55,228),(56,74),(56,113),(56,117),(56,125),(56,129),(56,156),(56,159),(56,223),(56,229),(57,73),(57,115),(57,119),(57,127),(57,131),(57,157),(57,159),(57,220),(57,226),(58,74),(58,114),(58,118),(58,126),(58,130),(58,157),(58,158),(58,221),(58,227),(59,75),(59,120),(59,121),(59,132),(59,133),(59,156),(59,157),(59,224),(59,230),(60,75),(60,122),(60,123),(60,134),(60,135),(60,158),(60,159),(60,225),(60,231),(61,84),(61,100),(61,184),(61,185),(61,196),(61,197),(61,297),(62,85),(62,101),(62,186),(62,187),(62,198),(62,199),(62,297),(63,87),(63,102),(63,188),(63,190),(63,201),(63,203),(63,296),(64,86),(64,103),(64,189),(64,191),(64,200),(64,202),(64,296),(65,76),(65,104),(65,185),(65,192),(65,200),(65,206),(65,292),(66,77),(66,105),(66,184),(66,192),(66,201),(66,207),(66,293),(67,78),(67,106),(67,187),(67,193),(67,202),(67,206),(67,295),(68,79),(68,107),(68,186),(68,193),(68,203),(68,207),(68,294),(69,80),(69,108),(69,188),(69,195),(69,199),(69,204),(69,292),(70,81),(70,109),(70,189),(70,194),(70,198),(70,204),(70,293),(71,82),(71,110),(71,190),(71,195),(71,197),(71,205),(71,295),(72,83),(72,111),(72,191),(72,194),(72,196),(72,205),(72,294),(73,240),(73,242),(73,246),(73,248),(73,337),(74,241),(74,243),(74,247),(74,249),(74,337),(75,244),(75,245),(75,250),(75,251),(75,337),(76,172),(76,317),(76,346),(76,352),(77,173),(77,318),(77,346),(77,353),(78,174),(78,320),(78,347),(78,352),(79,175),(79,319),(79,347),(79,353),(80,176),(80,317),(80,348),(80,351),(81,177),(81,318),(81,349),(81,351),(82,178),(82,320),(82,348),(82,350),(83,179),(83,319),(83,349),(83,350),(84,180),(84,321),(84,346),(84,350),(85,181),(85,321),(85,347),(85,351),(86,182),(86,322),(86,349),(86,352),(87,183),(87,322),(87,348),(87,353),(88,160),(88,284),(88,317),(88,364),(89,161),(89,284),(89,318),(89,365),(90,162),(90,285),(90,320),(90,364),(91,163),(91,285),(91,319),(91,365),(92,164),(92,286),(92,317),(92,363),(93,165),(93,287),(93,318),(93,363),(94,166),(94,286),(94,320),(94,362),(95,167),(95,287),(95,319),(95,362),(96,168),(96,284),(96,321),(96,362),(97,169),(97,285),(97,321),(97,363),(98,170),(98,287),(98,322),(98,364),(99,171),(99,286),(99,322),(99,365),(100,180),(100,260),(100,261),(100,272),(100,273),(100,316),(101,181),(101,262),(101,263),(101,274),(101,275),(101,316),(102,183),(102,264),(102,266),(102,277),(102,279),(102,315),(103,182),(103,265),(103,267),(103,276),(103,278),(103,315),(104,172),(104,261),(104,268),(104,276),(104,282),(104,311),(105,173),(105,260),(105,268),(105,277),(105,283),(105,312),(106,174),(106,263),(106,269),(106,278),(106,282),(106,314),(107,175),(107,262),(107,269),(107,279),(107,283),(107,313),(108,176),(108,264),(108,271),(108,275),(108,280),(108,311),(109,177),(109,265),(109,270),(109,274),(109,280),(109,312),(110,178),(110,266),(110,271),(110,273),(110,281),(110,314),(111,179),(111,267),(111,270),(111,272),(111,281),(111,313),(112,160),(112,172),(112,240),(112,252),(112,260),(112,340),(113,161),(113,173),(113,241),(113,252),(113,261),(113,341),(114,162),(114,174),(114,243),(114,253),(114,262),(114,340),(115,163),(115,175),(115,242),(115,253),(115,263),(115,341),(116,164),(116,176),(116,240),(116,254),(116,266),(116,339),(117,165),(117,177),(117,241),(117,255),(117,267),(117,339),(118,166),(118,178),(118,243),(118,254),(118,264),(118,338),(119,167),(119,179),(119,242),(119,255),(119,265),(119,338),(120,168),(120,180),(120,244),(120,252),(120,268),(120,338),(121,169),(121,181),(121,244),(121,253),(121,269),(121,339),(122,170),(122,182),(122,245),(122,255),(122,270),(122,340),(123,171),(123,183),(123,245),(123,254),(123,271),(123,341),(124,160),(124,176),(124,246),(124,256),(124,274),(124,344),(125,161),(125,177),(125,247),(125,256),(125,275),(125,345),(126,162),(126,178),(126,249),(126,257),(126,272),(126,344),(127,163),(127,179),(127,248),(127,257),(127,273),(127,345),(128,164),(128,172),(128,246),(128,258),(128,278),(128,343),(129,165),(129,173),(129,247),(129,259),(129,279),(129,343),(130,166),(130,174),(130,249),(130,258),(130,276),(130,342),(131,167),(131,175),(131,248),(131,259),(131,277),(131,342),(132,168),(132,181),(132,250),(132,256),(132,280),(132,342),(133,169),(133,180),(133,250),(133,257),(133,281),(133,343),(134,170),(134,183),(134,251),(134,259),(134,283),(134,344),(135,171),(135,182),(135,251),(135,258),(135,282),(135,345),(136,160),(136,161),(136,168),(136,311),(136,312),(136,316),(137,162),(137,163),(137,169),(137,313),(137,314),(137,316),(138,164),(138,166),(138,171),(138,311),(138,314),(138,315),(139,165),(139,167),(139,170),(139,312),(139,313),(139,315),(140,156),(140,232),(140,236),(140,304),(140,324),(140,328),(141,157),(141,233),(141,237),(141,304),(141,323),(141,327),(142,158),(142,234),(142,238),(142,304),(142,325),(142,329),(143,159),(143,235),(143,239),(143,304),(143,326),(143,330),(144,197),(144,201),(144,208),(144,220),(144,292),(144,323),(144,326),(145,196),(145,200),(145,209),(145,221),(145,293),(145,323),(145,325),(146,198),(146,202),(146,210),(146,222),(146,294),(146,324),(146,325),(147,199),(147,203),(147,211),(147,223),(147,295),(147,324),(147,326),(148,204),(148,205),(148,213),(148,224),(148,296),(148,323),(148,324),(149,206),(149,207),(149,212),(149,225),(149,297),(149,325),(149,326),(150,187),(150,189),(150,214),(150,226),(150,292),(150,327),(150,330),(151,186),(151,188),(151,215),(151,227),(151,293),(151,327),(151,329),(152,184),(152,190),(152,216),(152,228),(152,294),(152,328),(152,329),(153,185),(153,191),(153,217),(153,229),(153,295),(153,328),(153,330),(154,192),(154,193),(154,219),(154,230),(154,296),(154,327),(154,328),(155,194),(155,195),(155,218),(155,231),(155,297),(155,329),(155,330),(156,252),(156,256),(156,337),(156,339),(156,343),(157,253),(157,257),(157,337),(157,338),(157,342),(158,254),(158,258),(158,337),(158,340),(158,344),(159,255),(159,259),(159,337),(159,341),(159,345),(160,288),(160,331),(160,370),(161,288),(161,332),(161,371),(162,289),(162,334),(162,370),(163,289),(163,333),(163,371),(164,290),(164,331),(164,369),(165,291),(165,332),(165,369),(166,290),(166,334),(166,368),(167,291),(167,333),(167,368),(168,288),(168,335),(168,368),(169,289),(169,335),(169,369),(170,291),(170,336),(170,370),(171,290),(171,336),(171,371),(172,331),(172,354),(172,360),(173,332),(173,354),(173,361),(174,334),(174,355),(174,360),(175,333),(175,355),(175,361),(176,331),(176,356),(176,359),(177,332),(177,357),(177,359),(178,334),(178,356),(178,358),(179,333),(179,357),(179,358),(180,335),(180,354),(180,358),(181,335),(181,355),(181,359),(182,336),(182,357),(182,360),(183,336),(183,356),(183,361),(184,260),(184,298),(184,346),(184,364),(185,261),(185,299),(185,346),(185,365),(186,262),(186,301),(186,347),(186,364),(187,263),(187,300),(187,347),(187,365),(188,264),(188,301),(188,348),(188,362),(189,265),(189,300),(189,349),(189,362),(190,266),(190,298),(190,348),(190,363),(191,267),(191,299),(191,349),(191,363),(192,268),(192,302),(192,346),(192,362),(193,269),(193,302),(193,347),(193,363),(194,270),(194,303),(194,349),(194,364),(195,271),(195,303),(195,348),(195,365),(196,272),(196,299),(196,350),(196,364),(197,273),(197,298),(197,350),(197,365),(198,274),(198,300),(198,351),(198,364),(199,275),(199,301),(199,351),(199,365),(200,276),(200,299),(200,352),(200,362),(201,277),(201,298),(201,353),(201,362),(202,278),(202,300),(202,352),(202,363),(203,279),(203,301),(203,353),(203,363),(204,280),(204,303),(204,351),(204,362),(205,281),(205,303),(205,350),(205,363),(206,282),(206,302),(206,352),(206,365),(207,283),(207,302),(207,353),(207,364),(208,240),(208,298),(208,317),(208,367),(209,241),(209,299),(209,318),(209,367),(210,242),(210,300),(210,319),(210,367),(211,243),(211,301),(211,320),(211,367),(212,244),(212,302),(212,321),(212,367),(213,245),(213,303),(213,322),(213,367),(214,246),(214,300),(214,317),(214,366),(215,247),(215,301),(215,318),(215,366),(216,248),(216,298),(216,319),(216,366),(217,249),(217,299),(217,320),(217,366),(218,250),(218,303),(218,321),(218,366),(219,251),(219,302),(219,322),(219,366),(220,240),(220,273),(220,277),(220,311),(220,338),(220,341),(221,241),(221,272),(221,276),(221,312),(221,338),(221,340),(222,242),(222,274),(222,278),(222,313),(222,339),(222,340),(223,243),(223,275),(223,279),(223,314),(223,339),(223,341),(224,245),(224,280),(224,281),(224,315),(224,338),(224,339),(225,244),(225,282),(225,283),(225,316),(225,340),(225,341),(226,246),(226,263),(226,265),(226,311),(226,342),(226,345),(227,247),(227,262),(227,264),(227,312),(227,342),(227,344),(228,248),(228,260),(228,266),(228,313),(228,343),(228,344),(229,249),(229,261),(229,267),(229,314),(229,343),(229,345),(230,251),(230,268),(230,269),(230,315),(230,342),(230,343),(231,250),(231,270),(231,271),(231,316),(231,344),(231,345),(232,252),(232,284),(232,346),(232,367),(233,253),(233,285),(233,347),(233,367),(234,254),(234,286),(234,348),(234,367),(235,255),(235,287),(235,349),(235,367),(236,256),(236,284),(236,351),(236,366),(237,257),(237,285),(237,350),(237,366),(238,258),(238,286),(238,352),(238,366),(239,259),(239,287),(239,353),(239,366),(240,305),(240,331),(240,372),(241,306),(241,332),(241,372),(242,307),(242,333),(242,372),(243,308),(243,334),(243,372),(244,309),(244,335),(244,372),(245,310),(245,336),(245,372),(246,307),(246,331),(246,373),(247,308),(247,332),(247,373),(248,305),(248,333),(248,373),(249,306),(249,334),(249,373),(250,310),(250,335),(250,373),(251,309),(251,336),(251,373),(252,288),(252,354),(252,372),(253,289),(253,355),(253,372),(254,290),(254,356),(254,372),(255,291),(255,357),(255,372),(256,288),(256,359),(256,373),(257,289),(257,358),(257,373),(258,290),(258,360),(258,373),(259,291),(259,361),(259,373),(260,305),(260,354),(260,370),(261,306),(261,354),(261,371),(262,308),(262,355),(262,370),(263,307),(263,355),(263,371),(264,308),(264,356),(264,368),(265,307),(265,357),(265,368),(266,305),(266,356),(266,369),(267,306),(267,357),(267,369),(268,309),(268,354),(268,368),(269,309),(269,355),(269,369),(270,310),(270,357),(270,370),(271,310),(271,356),(271,371),(272,306),(272,358),(272,370),(273,305),(273,358),(273,371),(274,307),(274,359),(274,370),(275,308),(275,359),(275,371),(276,306),(276,360),(276,368),(277,305),(277,361),(277,368),(278,307),(278,360),(278,369),(279,308),(279,361),(279,369),(280,310),(280,359),(280,368),(281,310),(281,358),(281,369),(282,309),(282,360),(282,371),(283,309),(283,361),(283,370),(284,288),(284,374),(285,289),(285,374),(286,290),(286,374),(287,291),(287,374),(288,375),(289,375),(290,375),(291,375),(292,311),(292,317),(292,362),(292,365),(293,312),(293,318),(293,362),(293,364),(294,313),(294,319),(294,363),(294,364),(295,314),(295,320),(295,363),(295,365),(296,315),(296,322),(296,362),(296,363),(297,316),(297,321),(297,364),(297,365),(298,305),(298,374),(299,306),(299,374),(300,307),(300,374),(301,308),(301,374),(302,309),(302,374),(303,310),(303,374),(304,337),(304,366),(304,367),(305,375),(306,375),(307,375),(308,375),(309,375),(310,375),(311,331),(311,368),(311,371),(312,332),(312,368),(312,370),(313,333),(313,369),(313,370),(314,334),(314,369),(314,371),(315,336),(315,368),(315,369),(316,335),(316,370),(316,371),(317,331),(317,374),(318,332),(318,374),(319,333),(319,374),(320,334),(320,374),(321,335),(321,374),(322,336),(322,374),(323,338),(323,350),(323,362),(323,367),(324,339),(324,351),(324,363),(324,367),(325,340),(325,352),(325,364),(325,367),(326,341),(326,353),(326,365),(326,367),(327,342),(327,347),(327,362),(327,366),(328,343),(328,346),(328,363),(328,366),(329,344),(329,348),(329,364),(329,366),(330,345),(330,349),(330,365),(330,366),(331,375),(332,375),(333,375),(334,375),(335,375),(336,375),(337,372),(337,373),(338,358),(338,368),(338,372),(339,359),(339,369),(339,372),(340,360),(340,370),(340,372),(341,361),(341,371),(341,372),(342,355),(342,368),(342,373),(343,354),(343,369),(343,373),(344,356),(344,370),(344,373),(345,357),(345,371),(345,373),(346,354),(346,374),(347,355),(347,374),(348,356),(348,374),(349,357),(349,374),(350,358),(350,374),(351,359),(351,374),(352,360),(352,374),(353,361),(353,374),(354,375),(355,375),(356,375),(357,375),(358,375),(359,375),(360,375),(361,375),(362,368),(362,374),(363,369),(363,374),(364,370),(364,374),(365,371),(365,374),(366,373),(366,374),(367,372),(367,374),(368,375),(369,375),(370,375),(371,375),(372,375),(373,375),(374,375)],376)
 => ? = 2
[2,1,1,1,1,1] => [1,6] => ([(5,6)],7)
 => ([(0,1)],2)
 => 0
[3,1,1,1,1] => [1,1,5] => ([(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
 => 1

Description

The Frankl number of a lattice. For a lattice

$L$ on at least two elements, this is

$\max_x(|L|-2|[x, 1]|),$ where we maximize over all join irreducible elements and

$[x, 1]$ denotes the interval from

$x$ to the top element. Frankl's conjecture asserts that this number is non-negative, and zero if and only if

$L$ is a Boolean lattice.

The following 4 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000223The number of nestings in the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St001330The hat guessing number of a graph.